AN INTRODUCTION TO THE MEAN CURVATURE
FLOW
FRANCISCO MARTÍN AND JESÚS PÉREZ
Abstract. The purpose of these notes is to provide an introduction to those who want to learn more about geometric evolution
problems for hypersurfaces and especially those related to curvature flow. These diffusion problems lead to interesting systems of
nonlinear partial differential equations and provide the appropriate
mathematical modeling of physical processes.
Contents
1. Introduction
2
2. Existence y uniqueness
12
3. Evolution of the Geometry by the Mean Curvature Flow
18
4. A comparison principle for parabolic PDE’s
31
5. Graphical submanifolds. Comparison Principle and
Consequences
34
6. Area Estimates and Monotonicity Formulas
52
7. Some Remarks About Singularities
72
References
74
Date: July 18, 2014.
1991 Mathematics Subject Classification. Primary 53C44,53C21,53C42.
Key words and phrases. Mean curvature flow, singularities, monotonicity formula, area estimates, comparison principle.
Authors are partially supported by MICINN-FEDER grant no. MTM201122547.
1
2
FRANCISCO MARTIN AND JESUS PEREZ
1. Introduction
Mean Curvature Flow is an exciting and already classical mathematical research field. It is situated at the crossroads of several scientific disciplines: Geometric Analysis, Geometric Measure Theory, PDE’s Theory, Differential Topology, Mathematical Physics, Image Processing,
Computer-aided Design, among other. The purpose of these notes is to
provide an introduction to those who want to learn more about these
geometric evolution problems for curves and surfaces and especially
curvature flow problems. They lead to interesting systems of nonlinear
partial differential equations and provide the appropriate mathematical
modeling of physical processes such as material interface propagation,
fluid free boundary motion, crystal growth,...
In Physics, diffusion is known as a process which equilibrates spatial
variations in concentration. If we consider a initial concentration u0 on
a domain Ω ⊆ R2 and seek solutions of the linear heat equation
∂
u − ∆u = 0,
∂t
with initial data u0 and natural boundary conditions on ∂Ω, we obtain
a successively smoothed concentrations {ut }t>0 . When Ω = R2 , the
solutions to this parabolic PDE coincides with the convolution of the
initial data with the heat kernel (or Gaussian filter)
(1.1)
Φσ (x) =
1 −|x|2 /σ2
e
2πσ
with standard deviation sigma, i.e., ut2 /2 = Φt ∗ u0 (see Remark 6.18.)
In general, derivatives of ut are bounded for t > 0 in terms of bounds
on u0 . It follows that, even if you start with a heat distribution which
is discontinuous, it immediately becomes smooth. Moreover, solutions
converge smoothly (in C ∞ ) to constants as t → ∞ (eventual simplicity).
The heat equation has some surprising properties which carry over to
much more general parabolic equations.
The Maximum Principle: : At a point where ut attains a maximum in space (that is, in Ω), the second derivatives in each direction are non-positive. By the heat equation, the time derivative is non-positive. It follows that the maximum temperature,
umax (t) = supx∈Ω u(x, t), does not increase as time passes.
Gradient Flow: A further useful property which holds for many
but not all heat-type equations is the gradient property: The
MEAN CURVATURE FLOW
3
heat equation is the flow of steepest decrease of the Dirichlet
Energy:
Z
1
E(u) =
|(Du)(x)|2 dx.
2 Ω
Figure 1. A surface moving by mean curvature.
If we are interested in the smoothing of perturbed surface geometries,
it make sense to think in analogues strategies. So, the source of inspiration diffused throughout everything that follows is the classical heat
equation (1.1).
The geometrical counterpart of the Euclidean Laplace operator ∆ on
a smooth surface M 2 ⊂ R3 (or more generally, a hypersurface M n ⊂
Rn+1 ) is the Laplace-Beltrami operator, that we will denote as ∆M .
4
FRANCISCO MARTIN AND JESUS PEREZ
Thus, we obtain the geometric diffusion equation
∂
(1.2)
x = ∆Mt x,
∂t
for the coordinates x of the corresponding family of surfaces {Mt }t∈[0,T ) .
A classical formula by Weierstraß (see [DHKW92], for instance) says
that, given an orientable1 (hyper)surface in Euclidean space, one has:
~
∆Mt x = H,
~ means the mean curvature vector. This means that (1.2) can
where H
be written as:
∂
~
(1.3)
x(p, t) = H(p,
t)
∂t
The mean curvature
is known to be the first variation of the area
R
functional M 7→ M dµ (see [DHKW92,CM11,MIP12].) We will obtain
for the Area(Ω(t)) of a relatively compact Ω(t) ⊂ Mt that
Z
d
~ 2 dµt .
(Area(Ω(t)) = −
|H|
dt
Ω(t)
In other words, we get that the mean curvature flow is the corresponding gradient flow for the area functional:
The Mean Curvature Flow is the flow of steepest decrease
of surface area.
Moreover, we also have a nice maximum principle for this particular
diffusion equation.
Theorem (Maximum/Comparison principle). If two properly immersed
hypersurfaces of Rn+1 are initially disjoint, they remain so. Furthermore, embedded hypersurfaces remain embedded.
In this line of result, we would like to point out that:
• If the initial hypersurface M is convex (i.e., all the geodesic curvatures are positive, or equivalently M bounds a convex region
of Rn+1 ), then Mt is convex, for any t.
• If M is mean convex (H > 0), then Mt is also mean convex, for
any t.
1Throughout
are orientable.
these notes we shall always assume that the hypersurfaces of Rn+1
MEAN CURVATURE FLOW
5
Moreover, mean curvature flow has a property which is similar to the
eventual simplicity for the solutions of the heat equation. This result
was proved by Huisken and asserts:
Theorem. Convex, embedded, compact hypersurfaces converge to points
p ∈ Rn+1 . After rescaling to keep the area constant, they converge
smoothly to round spheres.
There is a rather general procedure for producing heat-like curvature flows. In general, we wish to evolve hypersurfaces M n in Rn+1
(or in a complete, Riemannian, (n + 1)-dimensional manifold). Then
any (smooth) symmetric function f of n variables, which is monotone
increasing in each variable, determines a suitable speed function:
F (p, t) := f (k1 (p, t), . . . , kn (p, t));
where ki , i = 1, . . . , n, represent the principal curvatures of Mt . This
yields a general class of curvature flows:
∂
(1.4)
x(p, t) = F (p, t) · ν(p, t),
∂t
where ν(·, t) is the Gauß map of Mt . Some of the most interesting
examples are:
(a) Mean Curvature Flow: f (x1 , . . . , xn ) =
n
X
xi , (F = H).
i=1
(b) Harmonic Mean Curvature Flow: f (x1 , . . . , xn ) =
(c) Gauß curvature flow: f (x1 , . . . , xn ) =
n
Y
n
X
1
x
i=1 i
!−1
.
xi , (F = K).
i=1
1
(d) Inverse Mean Curvature Flow: f (x1 , . . . , xn ) = − Pn
−H −1 ).
i=1
xi
, (F =
Applications of the mean curvature flow (and its variants: harmonic
mean curvature flow, inverse mean curvature flow,...) are numerous and
cover various aspects of Mathematics, Physics and Computing. In the
following paragraphs we will briefly describe some of these applications,
with particular emphasis on two of them. The inverse mean curvature
flow was used by Huisken and Ilmanen to prove the Riemann Penrose
inequality [HI01]. Similarly, Andrews got an alternative proof of the
topological version of the sphere theorem [And94] making use of the
harmonic mean curvature flow.
6
FRANCISCO MARTIN AND JESUS PEREZ
1.1. Riemannian Penrose Inequality. The Riemannian Penrose inequality is a special case of the unsettled Penrose Conjecture. In a
seminal paper [Pen73] (see also [Pen82]), in which he proposed the celebrated cosmic censorhip conjecture, R. Penrose also proposed a
related inequality, which today is know as “Penrose Inequality”. The
inequality is derived from cosmic censorship by using a heuristic argument relying on Hawking’s Area Theorem [HE73]. Consider a spacetime satisfying the so called dominant energy condition (DEC),
which contains an asymptotically flat Cauchy surface with ADM mass
m (see definition below), and containing an event horizon (roughly, the
area of a black hole) of area A = 4πr2 , which undergoes gravitational
collapse and settles to a Kerr-Newman solution of mass m∞ and area
radius r∞ . Physical arguments imply that the ADM mass of the final
state m∞ is no greater than m (no new mass appear, even though radiation may imply some loss of mass), then the area radius r∞ is no
less than r, and the final state must satisfy
1
m∞ ≥ r∞ .
2
The evolution of black holes (assuming that it is deterministic, i.e., no
naked singularity appears) implies that the area of its event horizon
must increase, so it must have been the case that
1
m ≥ r,
2
also at the beginning of the evolution.
A counterexample to the Penrose inequality would therefore suggests
data which leads under the Einstein evolution to naked singularities,
and a proof of the Penrose inequality may be viewed as evidence in
support of the cosmic censorship. The event horizon is indiscernible
in the original slice without knowing the full evolution, however one
may, without disturbing this inequality, replace the event horizon by
the (possible smaller) apparent horizon, the boundary of the region
admitting trapped surfaces. The inequality is even more simple in the
time-symmetric case, in which the apparent horizon coincides with the
outermost minimal surface, and the dominant energy condition reduces
to the condition of nonnegative scalar curvature. This leads to the
Riemannian Penrose inequality: the ADM mass m and the area radius
r of the outermost minimal surface in an asymptotically at 3-manifold
of nonnegative scalar curvature, satisfy
r
r
A
m≥ =
,
2
16π
MEAN CURVATURE FLOW
7
and the equality holds if and only if the manifold is isometric to the
canonical slice of the Schwarzschild spacetime. Note that this characterizes the canonical slice of Schwarzschild as the unique minimizer of
m among all such 3-manifolds admitting an outermost horizon of area
A.
For a more precise explanation about the physical interpretation of
the inequality we recommend [MS13]. In these notes, we will focus on
the mathematical aspects of the Riemannian Penrose inequality and
how the ICMF has been a key tool in their demonstration.
Consider (N 3 , g = (gij )) a Riemannian 3-manifold. Assume N is
asymptotically flat which means that:
(1) N is realized by an open set which is diffeomorphic to R3 \ K;
K compact,
C
(2) |gij − δij | ≤
, as |x| → ∞;
|x|
∂gij ≤ C , as |x| → ∞.
(3) ∂xk |x|2
(4) We also assume
Ric ≥ −
C
· g.
|x|2
In this setting we define
Definition 1.1 (Arnowitt-Deser-Misner (ADM) mass). The total energy, or ADM mass, of the end is defined by a flux integral through the
sphere at infinity:
Z
1 X
∂gij
∂gij
m := lim
−
· nj dµ,
r→+∞ 16π
∂x
∂x
2 (0,r)
j
i
S
i,j
where n represents the “outward” pointing Gauß map of the Euclidean
sphere S2 (0, r).
Although this flux is defined using local coordinates, it is global
invariant of the end.
Theorem 1.2 (Riemannian Penrose Inequality [HI01]). Let N be a
complete, connected 3-manifold (with boundary.) Suppose that
(a) N is asymptotically flat, with ADM mass m,
(b) N has nonnegative scalar curvature,
8
FRANCISCO MARTIN AND JESUS PEREZ
(c) N has compact boundary which consists of minimal surfaces, and
N contains no other compact minimal surfaces.
Then
r
Area(M )
,
16π
where M is any connected component of ∂N . Moreover, equality holds
iff N is isometric to one-half of the spatial Schwarzschild manifold.
m≥
The spatial Schwarzschild manifold is (R3 − {(0, 0, 0)}, g) where
4
m
g := 1 +
· g0 ,
2|x|
and g0 represents the Euclidean metric of R3 .
• It possesses an inversive isometry fixing S2 (0, m/2), which is an
area minimizing sphere of area 16πm2 .
• The manifold of the Riemannian Penrose inequality is R3 −
B(0, m/2).
Huisken-Ilmanen’s proof of the Riemannian Penrose inequality. Huisken and Ilmanen proved the inequality, including the rigidity
part, in [HI01] by using the inverse mean curvature flow, an approach proposed by Jang and Wald [JW77]. In this introduction, we
would like to give a rough idea of the structure of their proof. A classical solution of the Inverse Mean Curvature Flow (IMCF from now on)
is a smooth family of (hyper)surfaces F : M × [0, T ] → N, satisfying
the equation
∂
ν
(1.5)
F =− ,
∂t
H
where ν represents the Gauß map of Mt := F (M, t) and 0 < H(·, t)
is its mean curvature.
Without extra geometric hypotheses, the mean curvature could develop a zero and then the equation would present a singularity. That
was the reason because Huisken and Ilmanen introduced a level-set formulation of (1.5), where the evolving surfaces are given as level sets of
a scalar function φ:
Mt = ∂ ({x ∈ N / φ(x) < t}) ,
so (1.6) is replaced by the degenerate elliptic equation
MEAN CURVATURE FLOW
(1.6)
divN
Dφ
|Dφ|
9
= |Dφ|
They were able to overcome the problem that the evolving surface can
become singular before reaching infinity by formulating and analysing a
suitable weak notion of the solution of (1.6). These weak solutions are
locally Lipschitz continuous functions and the treatment was inspired
in a work of Evans and Spruck on the MCF [ES91].
It appears that neither (1.5) is a gradient flow nor (1.6) is an EulerLangrange equation. The idea of these two authors consists of freezing
|Dφ| in the right-hand side of (1.6) and consider (1.6) as the EulerLagrange equation of the functional:
Z
K
Jφ (ψ) = Jφ (ψ) :=
(|Dψ| + ψ|Dφ|) d µ,
K
where K is a compact subset of N .
Definition 1.3 (Weak solution). Let φ a locally Lipschitz function on
the open set Ω ⊆ N . Then we say that φ is a weakk solution of (1.6)
on Ω provided
JφK (φ) ≤ JφK (ψ),
for all ψ locally Lipschitz and such that {ψ 6= φ} ⊂⊂ Ω, where we are
integrating over any compact {ψ 6= φ} ⊆ K ⊂ Ω.
In order to understand the value of the above definition in the solution
of our problem, we need to introduce a new concept; the Hawking mass.
Given a compact surface M in N , the Hawking mass is defined by:
s
Z
Area(M )
2
16π −
H dµ .
mH (M ) :=
(16π)3
M
Hawking already noticed that mH approaches the ADM mass for large
coordinate spheres.
r
Area(M )
If M is minimal then mH (M ) is precisely
. Moreover, the
16π
Hawking mass has an especially nice behavior respect to the inverse
MCF:
I. Geroch Monotonicity Formula. Geroch [Ger73] introduced
the IMCF and realized that the mass mH of a family of surfaces
10
FRANCISCO MARTIN AND JESUS PEREZ
evolving by the inverse MCF is monotone nondecreasing, provided that the surface is connected and the scalar curvature of N
is nonnegative.
II. One of the main achievements of [HI01] consists of proving that
Geroch Monotonicity Formula also works for the Huisken-Ilmanen
weak solutions of the inverse MCF, even in the presence of jumps.
III. The derivative vanishes precisely on standard expanding spheres
in flat 3-space and Schwarzschild example.
Taking these properties into account the proof of the case of a connected
horizon (M = ∂N connected) works as follows. We move M by the
IMCF, obtaining a family of compact surfaces Mt which collapses at
the point of infinity. By the monotonicity formula, we know that
r
Area(M )
mH (Mt ) ≥ mH (M ) =
,
16π
(recall that M is minimal.) For t big enough, we have that mh (Mt )
approximates the ADM mass m, so
r
Area(M )
.
m ≈ mH (Mt ) ≥
16π
Moreover, the equality holds iff mH (Mt ) is constant along the flow, i.e.
its derivative vanishes. According to Property III, this only happens
for standard expanding spheres in Schwarzschild’s example.
Finally, we would like to mention that the inequality was proven in full
generality (non-connected horizon) by Bray [Bra01] using a conformal
flow of the initial Riemannian metric, and the positive mass theorem
[SY79].
1.2. The Sphere Theorem. It is known that if a manifold is simply
connected and has constant positive sectional curvatures, then it is a
sphere with the standard Riemannian metric.
• In the 1940’s, Heinrich Hopf asked whether we can also wiggle the geometry a little, instead only requiring that the sectional curvatures be close to some constant. Let’s say 1 − ε <
K ≤ 1.
• In 1951 Rauch [Rau51] proved a simply connected manifold with
curvature in [3/4, 1] is homeomorphic to a sphere.
• At the beginning of the 1960’s, Berger [Ber60] and Klingenberg [Kli61] confirmed the conjecture, with the optimal value
MEAN CURVATURE FLOW
11
of ε : A simply connected Riemannian manifold with sectional
curvatures in the interval (1/4, 1] is homeomorphic to a sphere.
If the value 1/4 is allowed, there are counterexamples. Actually, any compact symmetric space of rank 1 admits a metric
whose sectional curvatures lie in the interval [1, 4]. The list of
these spaces includes the following examples:
– The complex projective space CPk , for 2k ≥ 4.
– The quaternionic projective space HPk , for 4k ≥ 8.
– The projective plane over the octonions (dimension 16)
It remained an open conjecture for over 50 years that the conclusion of
homeomorphism should be improvable to diffemorphism. The problem
was solved in the affirmative by S. Brendle and R. Schoen in [BS09]
using the Ricci Flow.
However, as we mentioned before, the sphere theorem was also proved
by B. Andrews using curvature flows.
Idea of the proof: Using the pinching assumption, it is not difficult
to construct a large disk D(p, r) in M whose boundary is smooth and
convex in the “outwards” direction.
We would like to flow this boundary in the outwards direction to a
point via a suitable curvature flow. This would demonstrate that the
manifold is formed from gluing two disks together along their boundaries, and hence is a sphere.
We know that the mean curvature flow doesn’t work, but there’s at
least one flow speed that makes the job; namely, the harmonic mean
curvature:
!−1
n
X
∂
1
x = f · ν, f (k1 , . . . , kn ) =
.
∂t
k
i
i=1
Note that the conclusion in this case is stronger than homeomorphism: The manifold is diffeomorphic to a twisted sphere (two disks
glued by a diffeomorphism along their boundary). But this is still
slightly weaker than diffeomorphism.
1.3. Image processing. These ‘smoothing” properties that we mentioned of mean curvature flow make it an ideal tool in image processing,
computer aided geometric design and computer graphics. Here, issues are fairing, modeling, deformation, and motion. Constructive and
12
FRANCISCO MARTIN AND JESUS PEREZ
more explicit approaches based for instance on splines are nowadays
already classical tools. More recently geometric evolution problems
and variational approaches have entered this research field as well and
have turned out to be powerful tools. For those readers interested in
the applications of curvature flows in image processing we recommend
[CDR03].
Since our background is closer to Geometric Analysis, we have mainly
used the monographs [Eck04], [Man11] and [RS10] in the elaboration
of these notes. For readers who are more familiar with the language
and techniques of Geometric Measure Theory, we recommend [Bra78]
and [Ilm95].
These notes correspond to the contents of a mini-course given by
the first author at the Program on Geometry and Physics, Granada
2014. The authors are extremely grateful to all participants in this
program who have sent us corrections and suggestions that have helped
to improve these notes. In that sense, we feel especially indebted to
Miguel Sánchez for his valuable comments.
2. Existence y uniqueness
Along this section M will represent a n-dimensional submanifold of
Rn+1 . Although the most part of the results are also valid in a more
general setting, we will restrict our attention to the study of hypersurfaces in Euclidean space.
Definition 2.1. We say that M moves by the mean curvature if there
exists a smooth family of immersions F : M × [0, T ) → Rn+1 such that
• F (·, 0) is the original immersion of M in Rn+1 ;
• For each p in M and for each t in [0, T ) one has
∂
~
F (p, t) = H(p,
t),
∂t
~
where H(p,
t) is the mean curvature vector of F (M, t) at F (p, t).
Remark 2.2. We shall introduce the following notation. Given a map
F : M × [0, T ) → Rn+1 as in the previous definition, we will write
Ft := F (·, t). Thus, we will use the same notation for all the elements
MEAN CURVATURE FLOW
13
associated to the smooth family of immersions {Ft }t∈[0,T ) like, for in~ t (p) := H(p,
~
stance Mt := F (M, t), the mean curvature vector H
t), etc.
For simplicity in the computations, we will often suppress the subscript
t when no confusion is possible.
We will start with some local computations. Let (U, x1 , x2 , . . . , xn )
be local co-ordenates around p ∈ U ⊂ M n . We have F : (M n , g) →
(Rn+c , ḡ) where ḡ = h·, ·i is the usual scalar product in Rn+c and g =
dF ∗ (ḡ). So
∂
∂
∂
∂
∂F ∂F
gij = g(
,
) = ḡ dF
, dF
=
,
,
∂xi ∂xj
∂xi
∂xj
∂xi ∂xj
∂F 2
where we are writing dF ∂x∂ i = ∂x
.
i
Recall that we can locally identify M n with F (M n ). In this way,
∂F
we can say that ∂x
is a local vector field on Rn+k which extends the
i
smooth field on M n given by ∂x∂ i .
¯ in the Euclidean
On the other hand, the Levi-Civita connection ∇
n+c
n+c
space R
is the standard flat connection in R , i.e., given X̄, Ȳ ∈
n+c
¯ X̄ Ȳ = DX̄ Ȳ , where DX̄ Ȳ means the directional
X(R ) one has ∇
¯ ∂F ∂F , that
derivative of Y alongX. In what follows it will appear ∇
∂xj
we will denote as
is as follows:
∂2F
.
∂xi ∂xj
∂xi
It is an abuse of notation whose justification
2
¯ ∂F ∂F = D ∂F ∂F = D ∂ ∂F = ∂ F ,
∇
∂xi ∂x
∂xi ∂x
∂xi ∂x
∂xi ∂xj
j
j
j
where the second equality has taken into account the identification
∂F
between the fields ∂x∂ i and ∂x
given by F . We have also used that
i
¯
∇X̄ Ȳ = DX̄ Ȳ where D means the standard directional derivative in
Rn+c .
Recall that the mean curvature vector can be computed in terms of
the connection as follows3:
⊥
∂F
ij
¯ ∂F
~ = g ∇
H
.
∂xi ∂x
j
2We
X(U ) into X(F (U )), in such a way that
are considering dF as a map fromn+c
can be seen as a local field on R
.
3Given a vector v ∈ Rn+c and p ∈ M we can decompose v = v> + v⊥ where
⊥
v> ∈ Tp M and v⊥ ∈ (Tp M )
dF
∂
∂xi
14
FRANCISCO MARTIN AND JESUS PEREZ
And we can go further getting that:
⊥
>
∂F
∂F
∂F
ij ¯
ij ¯
ij ¯
g ∇ ∂F
= g ∇ ∂F
− g ∇ ∂F
=
∂xi ∂x
∂xi ∂x
∂xi ∂x
j
j
j
∂F
∂F
∂F ∂F
ij ¯
ij ¯
= g ∇ ∂F
− g ∇ ∂F
,
g kl
.
∂xi ∂x
∂xi ∂x
∂xl
j
j ∂xk
Proof. Indeed, we only have to check that
> ∂F
∂F ∂F
∂F
ij ¯
ij ¯
g ∇ ∂F
,
g kl
.
= g ∇ ∂F
∂xi ∂x
∂xi ∂x
∂xl
j
j ∂xk
To do this, fix m ∈ {1, 2, . . . , n}, then one has
∂F
∂F
∂F ∂F
∂F ∂F
ij ¯
kl ∂F
ij ¯
kl ∂F
,
,
,
,
g ∇ ∂F
g
= g ∇ ∂F
g
=
∂xi ∂x
∂xi ∂x
∂xl ∂xm
∂xl ∂xm
j ∂xk
j ∂xk
∂F ∂F
∂F ∂F
ij ¯
kl
ij ¯
k
= g ∇ ∂F
,
g glm = g ∇ ∂F
,
δm
=
∂xi ∂x
∂x
i ∂xj ∂xk
j ∂xk
>
∂F ∂F
∂F
∂F
ij ¯
ij ¯
,
=
g ∇ ∂F
,
.
= g ∇ ∂F
∂xi ∂x
∂xi ∂x
∂xm
j ∂xm
j
Using the non-degenerancy of the metric h·, ·i we complete the proof.
Hence, the Mean Curvature Flow equation
written in this new form:
∂
F (p, t)
∂t
~
= H(p,
t) can be
∂F
∂F ∂F
∂
∂F
ij ¯
ij ¯
F (p) = g ∇ ∂F
− g ∇ ∂F
,
g kl
,
∂xi ∂x
∂xi ∂x
∂t
∂xl
j
j ∂xk
¯ ∂F ∂F = ∂ 2 F we obtain
taking into account that ∇
∂xi ∂xj
∂xi ∂xj
2
2
∂
∂F
∂F
ij ∂ F
ij ∂ F
F (p) = g
− g
,
g kl
.
∂t
∂xi ∂xj
∂xi ∂xj ∂xk
∂xl
If we express the previous equation in coordinates we get:
n+k
2 α
X
∂ α
∂ 2 F β ∂F β ∂F α
ij ∂ F
ij kl
F (p) = g
−g g
,
∂t
∂xi ∂xj
∂xi ∂xj ∂xk ∂xl
β=1
where we clearly observe that we are dealing with a non-linear PDE.4
4The
second order coefficients are g ij , which depend on the map F .
MEAN CURVATURE FLOW
15
Another way of writing the MCF equation is the following:
⊥
>
∂F
∂F
∂F
ij ¯
ij ¯
ij ¯
~
H = g ∇ ∂F
− g ∇ ∂F
= g ∇ ∂F
=
∂xi ∂x
∂xi ∂x
∂xi ∂x
j
j
j
> 2
2
∂ F
∂ F
∂F
∂F
ij
ij
¯
(2.1) = g
− ∇ ∂F
=g
− ∇ ∂F
.
∂xi ∂x
∂xi ∂x
∂xi ∂xj
∂xi ∂xj
j
j
At this point, we would like to remind some notions related to the hessian and laplacian operators.
Definition 2.3. Let (M n , g) be a Riemannian manifold and let ∇ be
its Levi-Civita connection. The hessian of f ∈ C ∞ (M ) is defined as
the operator ∇2 f : X(M ) × X(M ) → C ∞ (M ) given by ∇2 f (X, Y ) :=
X(Y (f )) − (∇X Y )(f ) for any X, Y ∈ X(M ).
It is well known that ∇2 f is symmetric (to prove this we use that
∇ is torsion free) and C ∞ (M )-bilinear.
Definition 2.4. Given (M n , g) a Riemannian manifold and f ∈ C ∞ (M )
we define the Laplacian of f as ∆f := Trace(∇2 f ).
Notice that the map F : M n → Rn+c does not have a well defined
Laplacian. However, it makes sense to define the laplacian of each component F α . So, we define the laplacian of F as ∆F := (∆F 1 , ∆F 2 , . . . , ∆F n+c ).
Analogously, we can define also the hessian of F .
Bearing this in mind, we have that
∂F ∂F
∂F ∂F
∂F
2
∇F
,
=
(F ) − ∇ ∂F
(F ) =
∂xi ∂x
∂xi ∂xj
∂xi ∂xj
j
∂F
∂
∂
=
(F ) − ∇ ∂F
(F ) =
∂xi ∂x
∂xi ∂xj
j
∂ 2F
∂F
=
− ∇ ∂F
(F )
∂xi ∂x
∂xi ∂xj
j
and so equation (2.1) becomes
∂F
∂F
ij
2
~ = g · (∇ F )
H
,
= Trace(∇2 F ) = ∆F,
∂xi ∂xj
~ depends on the point p ∈ M and the
Remark 2.5. Recall that H
instant t. In particular, we want to emphasize the temporal dependence
of the mean curvature. Moreover, the intrinsic metric g also depends
16
FRANCISCO MARTIN AND JESUS PEREZ
on t. For this reason, it is more precise to denote the intrinsic laplacian
as ∆g(t) F .
Taking into account the above remark, one has:
~ = ∆g(t) F,
H
which provides a new way of writing the MCF equation:
(2.2)
∂F
= ∆g(t) F.
∂t
Using this expression we will prove the existence of the mean curvature flow for a small time period in the case of compact manifolds. In
the demonstration, we will use a technique known as “de Turck trick”.
As we shall see, this trick consists of reducing the original problem to
a strictly parabolic quasilinear problem.
Theorem 2.6 (Short Existence and Uniqueness). Let M be a
compact manifold and F0 : M → Rn+1 a given immersion. There exists
a positive constant T > 0 and a unique smooth family of immersions
F (·, t) : M → Rn+1 , t ∈ [0, T ), such that

∂
~
F (p, t) = H(p,
t) for all (p, t) ∈ M × [0, T ),
∂t

F (·, 0) = F0 .
Proof. First of all, notice that F0 = F (·, 0) is an immersion, so F (·, t)
also would be an immersion for t small enough (see, for instance, [GP74,
p. 35].) That means that we only take care about the existence and
solution of the above PDE.
Assume that for some vector field V = v k ∂x∂ k in M (the field V will
be fixed later) we have that the equation:
(2.3)
∂ F̃
∂ F̃
= ∆g(t) F̃ + v k
∂t
∂xk
has solution for initial data F0 , F̃ : M × [0, T ) → Rn+1 . We are going
to see that the same happens for the MCF equation with initial data
F0 .
Indeed, consider a family ϕt : M ×[0, T ) → M of diffeomorphisms of M .
Let Ft (p) := F̃t (ϕt (p)) = F̃ (ϕt (p), t), where F̃ is the aforementioned
MEAN CURVATURE FLOW
17
solution of (2.3) (for now, we are assuming that such a solution exists.)
t
Using the chain rule and (2.3) we compute ∂F
(p):
∂t
∂Ft
∂ F̃ (ϕt , t)
∂ F̃
∂ϕk
∂ F̃
(p) =
(p) =
(ϕt (p), t) t (p) +
(ϕt (p), t) =
∂t
∂t
∂xk
∂t
∂t
∂ϕk
∂ F̃
∂ F̃
(ϕt (p), t) t (p) + ∆g(t) F̃ (ϕt (p), t) + v k
(ϕt (p), t) =
=
∂xk
∂t
∂xk
∂ϕkt
∂ F̃
k
(ϕt (p), t) v +
(p) .
= ∆g(t) F̃ (ϕt (p), t) +
∂xk
∂t
Hence, to get a solution to the MCF equation it suffices to find a family
ϕt such that:
∂ϕt
= −V,
∂t
ϕ0 = id .
This is a initial value problem for a system of ODE’s and so we can find
a solution. Moreover, taking T > 0 small enough we can assume that
ϕt is a diffeomorphism, for any t ∈ [0, T ]. This is due to the fact that
the initial data is a diffeomorphism (the identity) and the fact that the
diffeomorphisms from a compact manifold into itself form a stable class
(see again [GP74, p. 35].)
Hence, Ft (p) = F̃ (ϕt (p), t) verifies:
∂Ft
(p) = ∆g(t) F̃ (ϕt (p), t) = ∆g(t) Ft (p),
∂t
F (p, 0) = F̃ (ϕ0 (p), 0) = F̃ (id(p), 0) = F̃ (p, 0) = F0 (p),
in other words, it represents a solution of the MCF equation with initial
data F0 .
Summarizing, we only have to see that (2.3) has a solution. To do
this we take the vector field V whose coordinates are given by v k :=
g ij (Γkij − (Γ0 )kij ), being Γkij the Christoffel symbols of M ((Γ0 )kij means
18
FRANCISCO MARTIN AND JESUS PEREZ
the Christoffel symbol at t = 0.) Then (2.3) becomes
∂ F̃
∂ F̃
= ∆g(t) F̃ + v k
∂t
∂xk
2
∂ F̃
∂ F̃
∂ F̃
ij
=g
− ∇ ∂ F̃
+ g ij (Γkij − (Γ0 )kij )
∂xi ∂xj
∂xi ∂xj
∂xk
2
∂ F̃
∂ F̃
∂ F̃
− Γkij
+ g ij (Γkij − (Γ0 )kij )
= g ij
∂xi ∂xj
∂xk
∂xk
2
∂ F̃
∂ F̃
= g ij
− (Γ0 )kij
.
∂xi ∂xj
∂xk
Thus, this particular choice of V implies that the equation:
∂ F̃
∂ F̃
= ∆g(t) F̃ + v k
∂t
∂xk
can be written (in coordinates) as follows:
∂ 2 F̃
∂ F̃
∂ F̃
= g ij
− g ij (Γ0 )kij
,
∂t
∂xi ∂xj
∂xk
which a system of quasilinear parabolic PDE’s because (g ij ) is a positive
definite matrix which only depends on the first derivatives of F . The
local theory of parabolic PDE’s [Tay96] and the fact that M is compact,
gives us the existence and uniqueness of the solution in a short interval
of time [0, T ). This concludes the proof.
3. Evolution of the Geometry by the Mean Curvature
Flow
In this section we study how the usual geometric quantities evolve
under the mean curvature flow.
Remark 3.1 (Notation). From now on, we will denote ∇i X = ∇
for any X ∈ X(M ).
For the sake of simplicity, we will often write:
∂f
for any function f ∈ C ∞ (M ),
∇i f =
∂xi
∂
∇i X =
X for any field X ∈ X(M ).
∂xi
∂
∂xi
X
MEAN CURVATURE FLOW
19
Theorem 3.2 (Evolution of the intrinsic geometry). The intrinsic metric and the volumen form evolve as follows:
(3.1)
∂
~ Aij i
gij = −2hH,
∂t
p
∂p
~ 2 det g,
det g = −|H|
∂t
∂
∂
where Aij := II ∂xi , ∂xj and II(·, ·) denotes the second fundamental
(3.2)
form of the corresponding immersion.
Proof. As all the computations are local, then we can assume that F
is an embedding.
Given a vector field X̄ in Rn+c we shall write:
¯ i X̄ := ∇
¯ ∂F X̄,
∇
∂xi
¯ t X̄ := ∇
¯ ∂F X̄.
∇
∂t
Using Schwarz’s formula, the MCF equation and the symmetry of the
Levi-Civita connection, one has
∂
∂ ∂F ∂F
∂F
∂F
∂F
∂F
¯t
¯t
,
,∇
gij =
= ∇
,
+
=
∂t
∂t ∂xi ∂xj
∂xi ∂xj
∂xi
∂xj
∂F
∂F ¯ ∂F
∂F
¯
, ∇j
,
+
=
= ∇i
∂t
∂xj
∂xi
∂t
∂F
∂F ¯ ~
¯
~
+
, ∇j (H) .
= ∇i (H),
∂xj
∂xi
∂F ∂F
~ is normal to M , then H,
~
As ∂x
is tangent to M and H
= 0. In
∂xj
j
particular,
∂
∂F
∂F
∂F
~
¯
~
~
¯
0=
H,
= ∇i (H),
+ H, ∇i
=
∂xi
∂xj
∂xj
∂xj
∂F
∂F
¯ i (H),
~
~ ∇
¯ ∂F
= ∇
+ H,
.
∂xi ∂x
∂xj
j
Therefore
∂F
∂F
¯
~
~
¯
= − H, ∇ ∂F
.
∇i (H),
∂xi ∂x
∂xj
j
20
FRANCISCO MARTIN AND JESUS PEREZ
And so, substituting in the previous equation, we obtain
∂
∂F
∂F ~
~
¯
¯
gij = − H, ∇ ∂F
− ∇ ∂F
,H =
∂xi ∂x
∂xj ∂x
∂t
j
i
⊥ ⊥ ∂F
∂F
~ =
~
¯
¯
= − H, ∇ ∂F
−
,H
∇ ∂F
∂xi ∂x
∂xj ∂x
j
i
∂
∂
∂
∂
~ =
~
= − H, II
,
− II
,
,H
∂xi ∂xj
∂xj ∂xi
~ Aij i − hAji , Hi
~ = −2hH,
~ Aij i,
= −hH,
where we have used that, by the symmetry of II: Aij = Aji .
√
Our next step is to study the evolution of det g 5; the volumen
form:
1
∂
∂p
det g = √
det g.
∂t
2 det g ∂t
∂
In order to compute ∂t
det g, we are going to use equation (3.1), Jacobi’s
formula for the derivative of a determinant:
∂
∂
ij
~ Akl i =
det g = Trace adj(g) g = Trace (det g)g · (−2)hH,
∂t
∂t
n
X
ij
~
~ Aij i =
= −2 det g Trace g · hH, Akl i = −2 det g
g ij hH,
i,j=1
X
n
∂
∂
ij
ij
~
~
g II(
= −2 det ghH,
g Aij i = −2 det g H,
,
) =
∂xi ∂xj
i,j=1
i,j=1
X
⊥ n
∂F
ij ¯
~
~ Hi
~ =
= −2 det g H,
g ∇ ∂F
= −2(det g)hH,
∂xi ∂x
j
i,j=1
n
X
~ 2,
= −2(det g)|H|
~ in local coordinates.
where we have used the expression of H
Therefore,
p
1
∂p
∂
1
~ 2 = −|H|
~ 2 det g.
det g = √
det g = √
(−2)(det g)|H|
∂t
2 det g ∂t
2 det g
5This
is a simplified notation, we should write
p
det gij (p, t).
MEAN CURVATURE FLOW
21
3.1. Evolution of the Extrinsic Geometry. Below we will study
the evolution of the extrinsic geometry. For simplicity assume that the
codimension is 1, since in higher codimension many of the quantities involved are tensors. The interested reader can find a detailed exposition
of the situation in higher codimension in [Smo11].
Let ν ∈ X⊥ (M ) be a unit normal field. We define the ν-second
fundamental form hν : X(M ) × X(M ) → C ∞ (M ) as the map given by
hν (X, Y ) := hν, II(X, Y )i.
Notice that hν is a bilinear, symmetric form. The self adjoint operator associated to hν is called the Weiengarten operator and it will be
denoted by Sν . Recall that hν (X, Y ) = hSν (X), Y i.
For a compact hypersurface in Rn+1 , there is a unique (up to the
sign) Gauß map. So, for the sake of simplicity, we will write h and S
instead of hν and Sν .
Fix p ∈ M and consider a set of normal coordinates around p. For
simplicity we are going to label Ei = ∂x∂ i . Then we know that Γkij (p) = 0
for all i, j, k (in particular , ∇Ei Ej (p) = Γkij (p)Ek (p) = 0). It is well
known that if we work with the Levi-Civita connection 6 we can assume
that gij (p) = δij . Finally, let us define
hij := h(Ei , Ej ).
At this point we are going to deduce the so called Simons’ identities,
that play an important role here and in other key aspects of the theory.
These are “Bochner-type” formulae relating the Laplacian of the second
fundamental form and the Hessian of mean curvature.
But first, we are going to get a local expression for the Laplace
operator in terms of the notation introduced in Remark 3.1. Given
f ∈ C ∞ (M ) and {Ei } a local basis of smooth tangent fields, then
∆f = Trace(∇2 f ) = g ij ∇2ij f = g ij (∇Ei ∇Ej f − ∇∇Ei Ej f ).
Given T is a symmetric 2-tensor on M , then T 2 is the symmetric
2-tensor defined (for codimension 1 sub manifolds) by:
X
T 2 (X, Y ) :=
T (X, Ei ) · T (Ei , Y ),
i
6To
get the existence of a set of normal coordinates we only need that the connection is symmetric, not necessarily torsion free.
22
FRANCISCO MARTIN AND JESUS PEREZ
where {Ei } is an arbitrary orthonormal frame. With this notation, we
have that:
Lemma 3.3 (Simons’ Identities). The following identities hold:
(3.3)
∆h = ∇2 H + H · h2 − |h|2 · h,
2
1
∆|h|2 = hh, ∇2 Hi + ∇h + H Trace(h3 ) − |h|4 ,
2
P
where hh, ∇2 Hi = i,j hij ∇2ij H
(3.4)
Proof. We are going to work locally around a point p ∈ M , with the
orthonormal frame {Ei } with ∇Ei Ej (p) = 0 and define
hij := h(Ei , Ej ).
Remark 3.4. To avoid confusion, “the squared of hij ” will be denoted
by (hij )2 , while the coefficients of the tensor h2 will be denoted as h2ij .
We will proceed similarly with h3 .
Recall that h2ij = hil g lm hmj , h3uv = huk g ki hil g lm hmv , and |h| =
(g ij g kl hik hjl )1/2 is the norm of the tensor h with respect to the metric
g.
First we prove ∆hij = ∇2ij H + Hh2ij − |h|2 hij .
As we are working in normal coordinates around p,
∆hij = g kl (∇Ek ∇El hij − ∇∇Ek El hij ) = δkl ∇Ek ∇El hij = ∇Ek ∇Ek hij .
In the abbreviated form we write ∆hij = ∇k ∇k hij .
On the other hand, from the Codazzi equation, we have
(3.5)
∇k hij = ∇i hkj .
Thus, we get ∆hij = ∇k ∇i hkj .
It is not hard to check that ∇k ∇i hkj = ∇i ∇k hkj + Rikjα hαk + Rikkβ hβj ,
where R(Ei , Ek , Ej , Eα ) = Rikjα and similarly for R̄. Taking into account that the ambience space is Euclidean, the Gauß equation gives:
0 − Rikjα = hAij , Akα i − hAiα , Akj i,
that is,
Rikjα = hAiα , Akj i − hAij , Akα i.
By definition, we have Aij = hij ν and taking into account that hν, νi =
1, then we obtain
Rikjα = hAiα , Akj i − hAij , Akα i =
= hiα hkj hν, νi − hij hkα hν, νi = hiα hkj − hij hkα .
MEAN CURVATURE FLOW
23
Analogously,
Rikkβ = hiβ hkk − hik hkβ .
Therefore
∆hij = ∇i ∇k hkj + Rikjα hαk + Rikkβ hβj =
= ∇i ∇k hkj + (hiα hkj − hij hkα )hαk + (hiβ hkk − hik hkβ )hβj =
= ∇i ∇j hkk + hiα hkj hαk − hij hkα hαk + hiβ hkk hβj − hik hkβ hβj .
In the last equality we have used the symmetry of h and (3.5):
∇k hkj = ∇k hjk = ∇j hkk .
Recalling that we are using normal coordinates around p, we deduce
H = g ij hij = δij hij = hii ,
|h|2 = g ij g kl hik hjl = δij δkl hik hjl = hik hik ,
h2ij = hil g lm hmj = hil δlm hmj = hil hlj ,
including all this facts in our previous computations, we get
∆hij = ∇i ∇j hkk + hiα h2αj − hij |h|2 + Hh2ij − hik h2kj =
= ∇i ∇j H + Hh2ij − |h|2 hij =
= ∇2ij H + Hh2ij − |h|2 hij ,
this concludes the proof of the first identity.
Now we want to prove that
2
1
∆|h|2 = hij ∇2ij H + ∇h + H Trace(h3 ) − |h|4 .
2
As we are working on normal coordinates we have that |h|2 = (hij )2 .
Then
1
1
∆|h|2 = ∆(hij )2 .
2
2
Moreover, we already showed that ∆(hij )2 = ∇k ∇k (hij )2 , therefore:
1
1
∆(hij )2 = ∇k ∇k (hij )2 .
2
2
Thus,
1
1
1
∆(hij )2 = ∇k ∇k (hij )2 = ∇k (2hij ∇k hij ) = (∇k hij )2 + hij ∇k ∇k hij =
2
2
2
= (∇k hij )2 + hij (∆hij )
= (∇k hij )2 + hij (∇2ij H + Hh2ij − |h|2 hij )
= (∇k hij )2 + hij ∇2ij H + Hhij h2ij − |h|4 .
24
FRANCISCO MARTIN AND JESUS PEREZ
where in the last two equalities we have used that:
∆hij = ∇k ∇k hij = ∇2ij H + Hh2ij − |h|2 hij ,
as we already checked in the proof of the previous identity.
Now, using the definition of trace of h3 and the fact that we are working
with normal coordinates around p, we get
Trace(h3 ) = g uv h3uv = δ uv h3uv = h3uu = huk g ki hil g lm hmu =
= huk δ ki hil δ lm hmu = huk hkl hlu .
On the other hand,
hij h2ij = hij (hil g lm hmj ) = hij (hil δ lm hmj ) = hij (hil hlj ) = hij hjl hli =
= huk hkl hlu ,
where we have used that h is symmetric.
Therefore, Trace(h3 ) = hij h2ij , which implies
1
∆|h|2 = (∇k hij )2 + hij ∇2ij H + H Trace(h3 ) − |h|4 .
2
2
Finally, let us check that (∇k hij )2 = ∇h . Using our local orthonor 2 P
mal frame{Ei } parallel at p, we have that ∇h = i,j |∇hij |2 . Then
2 X
∇h =
|∇hij |2
i,j
X
X
=
h∇hij , Er iEr ,
h∇hkl , Es iEs
i,j
k,l
X
X
h∇hkl , Es i hEr , Es i
=
h∇hij , Er i
i,j,r
k,l,s
X
X
=
h∇hij , Er i
h∇hij , Es i δrs
i,j,r
=
X
=
X
=
X
h∇hij , Er i
i,j,s
2
i,j,r
2
Er (hij )
i,j,r
2
∇r hij ,
i,j,r
Therefore,
2
1
∆|h|2 = hij ∇2ij H + ∇h + H Trace(h3 ) − |h|4 ,
2
MEAN CURVATURE FLOW
25
as we wanted to prove.
Theorem 3.5 (Evolution of the extrinsic geometry). In our setting we have:
(3.6)
∂
∂H ∂F ij
ν=−
g = −∇H
∂t
∂xi ∂xj
(3.7)
∂
hij = ∆hij − 2Hh2ij + |h|2 hij
∂t
(3.8)
∂
H = ∆H + |h|2 H
∂t
(3.9)
2
∂ 2
|h| = ∆|h|2 − 2∇h + 2|h|4 .
∂t
Proof. Along this proof, we will denote Ei =
∂F
.
∂xi
∂H ∂F ij
∂
ν = − ∂x
g .
i) Let us prove that ∂t
i ∂xj
∂
We start by decomposing the vector ∂t
ν in the base {Ei },
∂
∂
ij
ij
¯
ν=g
ν, Ei Ej = g ∇t ν, Ei Ej =
∂t
∂t
∂
ij
¯
hν, Ei i − ν, ∇t Ei
=g
Ej =
∂t
∂
¯ t Ei =
hν, Ei i = 0. Moreover, we know that ∇
ν ⊥ Ei , implies ∂t
∂F
∂F
¯t
¯i = ∇
¯ iH
~ =∇
¯ i H,
~ which implies
∇
=∇
∂xi
∂t
¯ i HiE
~ j=
= −g ij hν, ∇
∂
ij
~ − h∇
¯ i ν, Hi
~ Ej =
= −g
hν, Hi
∂xi
~ = hν, Hνi = Hhν, νi = H ·1 = H,
At this point, we use that hν, Hi
¯
~
¯
¯ i ν, νi = H · 0 = 0 and so:
and h∇i ν, Hi = h∇i ν, Hνi = Hh∇
= −g ij
∂H ∂F
.
∂xi ∂xj
∂
In order to prove that ∂t
ν = −∇H we only have to remember
∂f ∂
that, given a smooth function f then ∇f = g ij ∂x
. Applying
i ∂xj
26
FRANCISCO MARTIN AND JESUS PEREZ
the above expression to H we get:
∂
∂H ∂F ij
ν=−
g = −∇H.
∂t
∂xi ∂xj
ii) Let us prove now
∂
h
∂t ij
= ∆hij − 2Hh2ij + |h|2 hij .
∂
∂
∂
∂
∂ ¯
∂ ¯
⊥
hij =
II
,
, ν = h(∇
h∇i Ej , νi.
i Ej ) , νi =
∂t
∂t
∂xi ∂xj
∂t
∂t
By deriving, we get
∂ ¯
¯ t∇
¯ i Ej , νi + h∇
¯ i Ej , ∇
¯ t νi.
h∇i Ej , νi = h∇
∂t
On one hand, we have
¯ t∇
¯ i Ej = ∇
¯ i∇
¯ t Ej = ∇
¯ i∇
¯ t ∂F = ∇
¯ i∇
¯ jH
~ =
¯ i∇
¯ j ∂F = ∇
∇
∂xj
∂t
¯ i∇
¯ j (Hν).
=∇
On the other hand, using (3.6), we deduce
¯ t ν = ∂ ν = −g rs ∂H Es ,
∇
∂t
∂xr
which gives
∂ ¯
¯ t∇
¯ i Ej , νi + h∇
¯ i Ej , ∇
¯ t νi =
h∇i Ej , νi = h∇
∂t
¯ i∇
¯ j (Hν), νi + h∇
¯ i Ej , −g rs ∂H Es i.
= h∇
∂xr
Summarizing, we have
∂
∂ ¯
rs ∂H
¯ ¯
¯
hij = h∇
Es i.
i Ej , νi = h∇i ∇j (Hν), νi + h∇i Ej , −g
∂t
∂t
∂xr
¯ j (Hν) = (∇
¯ j H)ν +H(∇
¯ j ν). We can compute
Now, we deal with ∇
¯
∇j ν as we did in (3.6):
∂
∂
∂
lk
lk
¯
¯
∇j ν =
ν=g
ν, El Ek = g
hν, El i − hν, ∇j El i Ek =
∂xj
∂xj
∂xj
∂
∂
∂
lk
=g
0 − hν, II
,
i Ek = g lk (−hjl )Ek .
∂xj
∂xj ∂xl
Hence, we get
¯ j (Hν) = (∇
¯ j H)ν + H(∇
¯ j ν) = (∇
¯ j H)ν − Hg lk hjl Ek .
∇
MEAN CURVATURE FLOW
27
Moreover,
¯ i Ej , −g rs ∂H Es i = −g rs ∂H h∇
¯ i Ej , Es i = −g rs ∂H h(∇
¯ i Ej )> , Es i =
h∇
∂xr
∂xr
∂xr
∂H
∂H k
= −g rs
h∇i Ej , Es i = −g rs
hΓ Ek , Es i =
∂xr
∂xr ij
∂H k
∂H
= −g rs
Γij gks = −Γkij g rs gsk
=
∂xr
∂xr
∂H
∂H
= −Γkij δrk
= −Γkij
.
∂xr
∂xk
Therefore,
¯ i∇
¯ j (Hν), νi + h∇
¯ i Ej , −g rs ∂H Es i =
h∇
∂xr
¯ i (∇
¯ j H)ν − Hg lk hjl Ek , νi − Γkij ∂H .
= h∇
∂xk
At this point, we have that:
∂
¯ i (∇
¯ j H)ν − Hg lk hjl Ek , νi − Γkij ∂H .
hij = h∇
∂t
∂xk
Notice that one has
¯ i (∇
¯ j H)ν − (Hg lk hjl )Ek = ∇
¯ i (∇
¯ j H)ν − ∇
¯ i (Hg lk hjl )Ek =
∇
¯ i (Hg lk hjl ) Ek − (Hg lk hjl )∇
¯ i Ek .
¯ i (∇
¯ j H) ν + (∇
¯ j H)∇
¯ iν − ∇
= ∇
We make the scalar product with ν, taking into account that
∂
∂
⊥
¯
¯
h∇i Ek , νi = h(∇i Ek ) , νi = II
,
, ν = hik ,
∂xi ∂xk
and we get
¯ i (∇
¯ j H)ν − Hg lk hjl Ek , νi = ∇
¯ i (∇
¯ j H) − Hg lk hjl hik .
h∇
In short, we proved that
∂
¯ i (∇
¯ j H) − Hg lk hjl hik − Γk ∂H .
hij = ∇
ij
∂t
∂xk
Now observe that by definition of Hessian
of a differentiable func
tion we have:∇2 f (X, Y ) = X Y (f ) − (∇X Y )(f ). In particular, for f = H, X = ∂x∂ i , Y = ∂x∂ j , and taking into account
∂f
∇ ∂ ∂x∂ j = Γkij and using the notation ∂x
= ∇i f , one obtains
i
∂xi
that
∂
∂
k
2
¯
¯
∇i ∇j H − Γij ∇k H = ∇ H
,
= ∇2ij H.
∂xi ∂xj
28
FRANCISCO MARTIN AND JESUS PEREZ
As g y h are symmetric, g lk hjl hik = hik g kl hlj = h2ij .
In short,
∂
hij = ∇2ij H − Hh2ij .
∂t
Finally, using Simons’ inequality (3.3) one gets
∇2ij H − Hh2ij = ∆hij − 2Hh2ij + |h|2 hij ,
which concludes the proof of (3.7).
∂
iii) Now we prove ∂t
H = ∆H + |h|2 H.
This time, we will use that the mean curvature can be also computed like H = g ij hij .
∂ ij
∂ ij
∂
∂
ij
H = g hij =
g hij + g
hij .
∂t
∂t
∂t
∂t
As we saw in the proof of the previous item,
∂
hij = ∆hij − 2Hh2ij + |h|2 hij .
∂t
∂ ij
To compute ∂t g recall that we knew, from (3.1), that
∂
~ Aij i = −2hHν, Aij i = −2Hhν, Aij i = −2Hhij .
gij = −2hH,
∂t
As g ij = g il glk g kj 7, then
∂
∂ ij
il
kj
g =
g glk g
=
∂t
∂t
∂ il
∂
∂ kj
kj
il
kj
il
g glk g + g
glk g + g glk
g
=
=
∂t
∂t
∂t
∂ il j
∂
∂ kj
il
kj
i
=
g δl + g
glk g + δk
g
=
∂t
∂t
∂t
∂
∂
∂ ij
il
= g +g
glk g kj + g ij =
∂t
∂t
∂t
∂
∂
glk g kj ,
= 2 g ij + g il
∂t
∂t
so substituting,
∂ ij
∂
il
g = −g
glk g kj = −g il (−2Hhij )g kj = 2Hg il hij g kj =
∂t
∂t
= 2Hhik g kj = 2Hhij .
7Indeed,
g il glk g kj = δki g kj = g ij .
MEAN CURVATURE FLOW
29
Therefore,
∂
∂
∂ ij
ij
H=
g hij + g
hij =
∂t
∂t
∂t
= 2Hhij hij + g ij (∆hij − 2Hh2ij + |h|2 hij ).
Simons’ identity (3.3) implies ∆hij = ∇2ij H + Hh2ij − |h|2 hij , land
so ∆hij − 2Hh2ij + |h|2 hij = ∇2ij H − |h|2 hij . Then,
∂
H = 2Hhij hij + g ij (∇2ij H − |h|2 hij )
∂t
= 2Hhij hij + g ij ∇2ij H − |h|2 g ij hij
= 2Hhij hij + Trace(∇2 H) − |h|2 H
= ∆H + H(2hij hij − |h|2 ).
Note that
|h|2 = g ij g kl hik hjl = g ij g lk hki hjl = g ji hli hjl = hlj hjl = hjl hjl = hij hij ,
therefore
2hij hij − |h|2 = 2|h|2 − |h|2 = |h|2 .
In short
∂
H = ∆H + |h|2 H.
∂t
2
∂
iv) Finally, we will prove ∂t
|h|2 = ∆|h|2 − 2∇|h| + 2|h|4 .
By definition, we have
|h|2 = g ij g kl hik hjl .
From the proof of the former item, we already know
∂ ij
g = 2Hhij ,
∂t
∂
hij = ∆hij − 2Hh2ij + |h|2 hij
∂t
(it is precisely (3.7)).
30
FRANCISCO MARTIN AND JESUS PEREZ
Then
∂ 2
∂
|h| = (g ij g kl hik hjl ) =
∂t
∂t
∂ ij kl
∂ kl
ij
=
g g hik hjl + g
g hik hjl +
∂t
∂t
∂
∂
ij kl
ij kl
hik hjl + g g hik
hjl =
g g
∂t
∂t
= 2Hhij g kl hik hjl + g ij 2Hhkl hik hjl +
g ij g kl (∆hik − 2Hh2ik + |h|2 hik )hjl +
g ij g kl hik (∆hjl − 2Hh2jl + |h|2 hjl ).
If in this last expression we rename the indices of the second term
by changing i for k, j for l, k for i and l for j, and in the fourth
addend changing i to j, j for i, k for l and l for k, then you get
∂ 2
|h| = 4Hhij g kl hik hjl + 2g ij g kl (∆hik − 2Hh2ik + |h|2 hik )hjl .
∂t
In the proof of Lemma 3.3 we got
h2ij = g kl hik hjl ,
hij h2ij = Trace(h3 ),
therefore
4Hhij g kl hik hjl = 4H Trace(h3 ).
Using (3.3),
∆hik − 2Hh2ik + |h|2 hik = ∇2ik H − Hh2ik .
So we get
∂ 2
|h| = 4H Trace(h3 ) + 2g ij g kl (∇2ik H − Hh2ik )hjl .
∂t
It is sufficient to check the equality at a point p in which it can
be assumed that are considered normal coordinates (in particular,
g ij (p) = δij and g kl (p) = δkl ), thus the last addend of the above
expression can be simplified as follows
2g ij g kl (∇2ik H − Hh2ik )hjl = 2hik (∇2ik H − Hh2ik ) =
= 2hik ∇2ik H − 2Hhik h2ik = 2hik ∇2ik H − 2H Trace h3 .
So far, we have proven
(3.10)
∂ 2
|h| = 2H Trace(h3 ) + 2hik ∇2ik H.
∂t
MEAN CURVATURE FLOW
31
At this point recall that we want to see
2
∂ 2
|h| = ∆|h|2 − 2∇h + 2|h|4 .
∂t
According to Simons’ identity (3.4),
2
∆|h|2 = 2hij ∇2ij H + 2∇h + 2H Trace(h3 ) − 2|h|4 ,
thereby substituting the expression we want to show, we get
∂ 2
|h| = 2hij ∇2ij H + 2H Trace(h3 ).
∂t
Comparing (3.10) and (3.11) we see that both coincide. This completes the proof.
(3.11)
4. A comparison principle for parabolic PDE’s
As we have seen in section 2, understanding and working with the
mean curvature flow involves a good knowledge about parabolic partial
differential equations. As it is well known, these equations generally
can not be solved, forcing us to look for results about the qualitative
behavior of the solutions. In this section we prove a theorem in that
direction which is known as the principle of comparison. Throughout
this section we will use Lieberman’s book [Lie96].
Let Ω be a domain (open, connected subset) in Rn+1 . In this setting
we will write the points of Rn+1 as X = (x, t), where x ∈ Rn .
Definition 4.1 (Parabolic Boundary). We define the parabolic boundary of Ω, denoted as ∂P Ω, as the set of points X = (x, t) ∈ ∂Ω (that
is, the topological boundary of Ω) such that for all > 0 the parabolic
cylinder Q(X, ) contains points in the complement of Ω; the definition
of parabolic cylinder Q(X0 , ) is
Q(X0 , ) := {Y ∈ Rn+1 : |Y − X0 | < , t < t0 },
p
where X0 = (x0 , t0 ) and |(x, t)| = max{|x|Rn , |t|}.
In the simplest case Ω = D × (0, T ), D a domain in Rn and T > 0,
we have that the parabolic boundary of Ω coincides with ∂P Ω = BΩ ∪
SΩ ∪ CΩ, where BΩ := Ω × {0} ( the “bottom” Ω), SΩ := ∂Ω × (0, T )
(the “side” of Ω) and CΩ := ∂(Ω) × {0} (the “corner” of Ω.)
32
FRANCISCO MARTIN AND JESUS PEREZ
Given u ∈ C 2,1 (Ω) we define the quasi-linear, second-order operator
P as
∂u
2
(4.1)
P u := −
+ aij (X, u, Du)Dij
u + a(X, u, Du).
∂t
We assume that aij (X, z, p) y a(X, z, p) are defined for any (X, z, p) ∈
Ω × R × Rn .
We say that P is parabolic in a subset S of Ω × R × Rn if the matrix
whose coefficients are aij (X, z, p) is positive definite for any (X, z, p) ∈
S. We distinguish two especial cases for S:
• If S = Ω × R × Rn , we say that P is parabolic;
• If S = {(X, z, p) ∈ U × R × Rn : z = u(X), p = Du(X)} for
some function u ∈ C 1 (U ) where U ⊂ Ω, then we will say that
P is parabolic at u.
Lemma 4.2. Let A and B symmetric real matrices of order n, with A
positive definite and B negative semidefinite. Then Trace(A · B) ≤ 0.
Proof. As A is positive definite, then A has a squared root, that we
denote by A1/2 , which is regular and symmetric. By Sylvester’s law of
inertia, as B and A1/2 B(A1/2 )> = A1/2 BA1/2 are congruent symmetric
matrices8, then they have the same number of positive, negative and
null eigenvalues. This means that all the eigenvalues µi of A1/2 BA1/2
are non-positive. Then, we have
n
X
0≥
µi = Trace(A1/2 BA1/2 ) = Trace((A1/2 B)A1/2 ) =
i=1
= Trace(A1/2 (A1/2 B)) = Trace(AB)
Theorem 4.3 (Lieberman, Comparison Principle). Let P be a
quasi-linear operator like in (4.1). Suppose aij (X, z, p) dos not depend
on z and that there exists a positive increasing function K(L) such that
a(X, z, p) − K(L) · z is decreasing in z on Ω × [−L, L] × Rn for L > 0.
If u and v are functions in C 2,1 (Ω \ ∂P Ω) ∩ C(Ω) such P is parabolic
at either u or v, P u ≥ P v on Ω \ ∂P Ω and u ≤ v on ∂P Ω, then u ≤ v
on Ω.
Proof. First, we fix L in such a way that [−L, L] contains the range of
u and v , i.e., L := max{supΩ |u|, supΩ |v|}.
8Notice
that B = P > A1/2 BA1/2 P ; where P = (A1/2 )−1 .
MEAN CURVATURE FLOW
33
Let us define w := (u − v)eλt en Ω, where λ is a real constant to be
determine later. Notice that u ≤ v on Ω is equivalent to prove that
w ≤ 0 on Ω. From our assumptions, we know that u ≤ v on ∂P Ω, so
we have to prove that:
w ≤ 0 en Ω \ ∂P Ω.
So would finish if we see that w can not have a positive interior maximum. We proceed by contradiction: Suppose that X0 = (x0 , t0 ) is
an interior maximum such that w(X0 ) > 0. Classical Analysis says to
us that if a function of class C 2 reaches the maximum at an interior
point, then the gradient at that point is zero and the Hessian matrix
is negative semidefinite 9. In particular, we have:
(4.2) Dw(X0 ) = 0 ⇔ (Du − Dv)(X0 )eλt0 = 0 ⇒ Du(X0 ) = Dv(X0 ),
∂w
∂
(X0 ) = 0 ⇔ (u − v)(X0 )eλt0 + λ(u − v)(X0 )eλt0 = 0,
∂t
∂t
which implies:
(4.3)
(4.4)
λ(u − v)(X0 ) = −
∂
(u − v)(X0 ),
∂t
The matrix (Dij w(X0 ))i,j=1,...,n is negative semidefinite10.
Below, for convenience, we denote R := (X0 , u(X0 ), Du(X0 )) and S :=
(X0 , v(X0 ), Dv(X0 )). From our hypothesis we know P u ≥ P v on Ω \
∂P Ω; in particular
0 ≤ P u(X0 ) − P v(X0 ) =
using the definition of the operator P
∂
(u − v)(X0 ) =
∂t
Notice that the first two terms aij (R) = aij (X0 , u(X0 ), Du(X0 )) and
aij (S) = aij (X0 , v(X0 ), Dv(X0 )) coincide. This is due to the fact that
= aij (R)Dij u(X0 ) − aij (S)Dij v(X0 ) + a(R) − a(S) −
9We
are assuming that n ≥ 2. If n = 1 would have that the derivative vanishes
at that point and the second derivative is positive at that point. We no longer do
this distinction, but it is clear that what follows is valid in the case n = 1 with the
obvious changes.
10If X is an interior maximum of w, then the Hessian matrix w in X , Hw(X ) =
0
0
0
(Dij w(X0 ))i,j=1,...,n,t , is negative semidefinite. Therefore, by the criterion of the
principal minors, the square submatrix obtained by removing the last row and last
column of Hw(X0 ), which is (Dij w(X0 ))i,j=1,...,n , remains a negative semidefinite
matrix.
34
FRANCISCO MARTIN AND JESUS PEREZ
aij does not depend on the second argument and because, by(4.2),
Du(X0 ) = Dv(X0 ). Using the above fact and (4.3) we have:
= aij (R)Dij (u − v)(X0 ) + a(R) − a(S) + λ(u − v)(X0 ) ≤
Applying Lemma 4.2 to the matrices A := (aij (R))i,j=1,...,n and B :=
(Dji (u − v)(X0 ))i,j=1,...,n , we get aij (R)Dij (u − v)(X0 ) ≤ 0, and so
≤ a(R) − a(S) + λ(u − v)(X0 ) ≤
At this point, recall that w(X0 ) > 0 , i.e., u(X0 ) > v(X0 ). As
a(x, z, p) − K(L)z is a decreasing function of z in Ω × [−L, L] × Rn ,
then a(R) − K(L)u(X0 ) ≤ a(S) − K(L)v(X0 ), or in other words,
a(R) − a(S) ≤ K(L)(u − v)(X0 ),
≤ K(L)(u − v)(X0 ) + λ(u − v)(X0 ) ≤ [K(L) + λ](u − v)(X0 ).
Now, It suffices to take λ < −K(L) to get that (u − v)(X0 ) ≤ 0, that
is, w(X0 ) ≤ 0, which is contrary to w(X0 ) > 0. This contradiction
proves the theorem.
5. Graphical submanifolds. Comparison Principle and
Consequences
The mean curvature flow has been extensively studied in some families
which have specific geometric conditions, as is the case of hypersurfaces,
Lagrangian submanifolds, graphs, etc. In this section we will focus on
the study of smooth graphs.
Let u : Rn → R be a smooth function. It is well known that the graph
of u, Graph(u) = {(x, u(x)) : x ∈ Rn } is a hypersurface Rn+1 . Then,
we can study how it evolves under mean curvature flow. The first goal
of this section will be to deduce the evolution equation for ut . Next,
we will see that we can apply a comparison principle to obtain a result
known as avoidance principle.
We start with a graph M0 = {(x, u(x)) : x ∈ Rn } ⊂ Rn+1 . We are
looking for a map
F : Rn × [0, T ) → Rn+1
F (p, t) = (x(p, t), u(x(p, t), t))
satisfying the MCF equation
∂F
= Hν,
∂t
MEAN CURVATURE FLOW
35
and such that F (·, 0) = M0 = Graph(u). At this point, it would be extremely useful to know how to express the main geometrical quantities
associated to M0 in terms of u and its derivatives. This is the purpose
of the next lemma
Lemma 5.1 (Graphical submanifolds). Let u : Rn → R be a smooth
function and M0 = Graph(u). Then, we have:
(1) gij = δij + Di uDj u,
Di uDj u
(2) g ij = δij −
,
1 + |Du|2
(−Du, 1)
,
(3) ν = p
1 + |Du|2
2
u
Dij
,
(4) hij = p
1+ |Du|2
Du
(5) H = div p
.
1 + |Du|2
Proof.
(1) The proof of the first item is straightforward:
gij = hDi F, Dj F i = h(ei , Di u), (ej , Dj u)i = δij + Di uDj u,
where {e1 , . . . en+1 } denotes the canonical basis of Rn+1 .
(2) If we make the matrix product, we get
gik g kj = (δik + Di uDk u)(δkj −
Dk uDj u
)=
1 + |Du|2
Dk uDj u
Dk uDj u
+ Di uDk uδkj − Di uDk u
=
2
1 + |Du|
1 + |Du|2
Di uDj u
Di uDj u
= δij −
+ Di uDj u − |Du|2
=
2
1 + |Du|
1 + |Du|2
Di uDj u
= δij − (1 + |Du|2 )
+ Di uDj u =
1 + |Du|2
= δij − Di uDj u + Di uDj u = δij ,
= δik δkj − δik
which means that (g ij ) is the inverse matrix of (gij ).
(3) The vectors {Di F = (ei , Di u), i = 1, . . . n} are a global basis
of the tangent bundle of Graph(u).Then (−Du, 1) is a normal,
36
FRANCISCO MARTIN AND JESUS PEREZ
non-vanishing vector field. So, we only have to normalized it
(−Du, 1)
ν=p
.
1 + |Du|2
(4)
¯ D F Dj F )⊥ , νi = h∇
¯ D F Dj F, νi =
hij = hII(Di F, Dj F ), νi = h(∇
i
i
(−Du , 1)
2
2
= hDDi F Dj F, νi = hDij F, νi = (0, Dij u), p
=
1 + |Du|2
2
Dij
u
=p
.
1 + |Du|2
(5) On one hand, we have
Du
Di u
div p
= Di p
=
1 + |Du|2
1 + |Du|2
Pn
D2 u
1
1
2hDi (Du), Dui =
= p i=1 ii − Di u
(1 + |Du|2 )3/2
1 + |Du|2 2
=p
=p
(5.1)
∆u
1 + |Du|2
∆u
1 + |Du|2
− Di u
−
1
hD2 uDj u, Dui =
(1 + |Du|2 )3/2 ji
2
2
uDj u
u
Di uDji
Di uDj uDij
∆u
p
−
=
.
2
3/2
2
3/2
(1 + |Du| )
1 + |Du|2 (1 + |Du| )
We number this auxiliary result that we will use it a few times
in later calculations:
2
Di uDj uDij
u
Du
∆u
.
−
div p
=p
2
3/2
1 + |Du|2
1 + |Du|2 (1 + |Du| )
On the other hand,
2
Dij
u
Di uDj u
ij
p
H = g hij = δij −
=
1 + |Du|2
1 + |Du|2
2
2
2
Di uDj uDij
u
Di uDj uDij
u
Dij
u
∆u
p
p
−
−
= δij
=
.
1 + |Du|2 (1 + |Du|2 )3/2
1 + |Du|2 (1 + |Du|2 )3/2
Therefore,
H = div
Du
p
1 + |Du|2
.
MEAN CURVATURE FLOW
37
Now we use this information to find new partial differential equations
for graphs that evolve by mean curvature. We start by deriving with
respect to time the application F . Deriving component by component
and applying the chain rule,
∂F
∂x ∂u ∂xi ∂u
∂x
∂x
∂u
=
,
+
=
, Du,
+
.
∂t
∂t ∂xi ∂t
∂t
∂t
∂t
∂t
Using this, the equation of the mean curvature flow becomes:
∂x
∂x
∂u
H
, Du,
+
=p
(−Du, 1).
∂t
∂t
∂t
1 + |Du|2
or equivalently
(5.2)
(5.3)

Du
∂x



 ∂t = −H p1 + |Du|2 ,

∂u
H
∂x


.
 Du, ∂t + ∂t = p
1 + |Du|2
Notice that, using 5.2, one has
∂x
H
H
Du,
= −p
hDu, Dui = − p
|Du|2 .
2
2
∂t
1 + |Du|
1 + |Du|
Therefore, equation (5.3) can be written as follows:
p
∂u
1
= H 1 + |Du|2
= H(1 + |Du|2 ) p
∂t
1 + |Du|2
Du
Finally, substituting H = div √
in (5.2) and (5.4) they be1+|Du|2
come:


∂x
Du
Du


p
p
=
−
div
·
,
(5.5)

 ∂t
1 + |Du|2
1 + |Du|2
p

∂u
Du


p
=
div
·
1 + |Du|2 .
(5.6)

 ∂t
2
1 + |Du|
(5.4)
We need some extra computations to get an expression for the above
divergence. According to (5.1),
Pn
2
2
Di uDj uDij
u
Du
i=1 Dii u
p
p
div
=
−
.
1 + |Du|2
1 + |Du|2 (1 + |Du|2 )3/2
38
FRANCISCO MARTIN AND JESUS PEREZ
Then,
Pn
2
u
Di uDj uDij
D2 u
div p
= p i=1 ii −
=
2
3/2
1 + |Du|2
1 + |Du|2 (1 + |Du| )
2
uDj u
Di uDji
1
2
δij Dij u −
=
=p
1 + |Du|2
1 + |Du|2
1
Di uDj u
2
=p
δij −
Dij
u.
2
2
1
+
|Du|
1 + |Du|
Du
Substituting in (5.6),
∂u
Di uDj u
D2 u(Du, Du)
2
= δij −
,
(5.7)
D
u
=
∆u
−
ij
∂t
1 + |Du|2
1 + |Du|2
where recall that D2 u means the Hessian operator associated to u.
With all we have seen so far we can prove the following well-known
result for mean curvature flow of a sphere.
Proposition 5.2. The spheres of Euclidean space evolve under the
mean curvature flow as spheres that concentrically contract until collapse in finite time at one point; the common center of the family of
spheres.
Proof. The upper n-dimensional hemisphere of radius ρ canpbe seen as
the graph of the function u : B(0, ρ) → R given by u(x) := ρ2 − |x|2 ,
where B(0, ρ) ⊂ Rn means the Euclidean ball centered at the origin of
radius ρ.
The abundance of symmetry of the sphere will allow us to solve the
partial differential equation of the mean curvature flow in this particular
case. Indeed,
xi
Di u = − p
,
ρ2 − |x|2
and from here we get
Du = − p
x
ρ2 − |x|2
,
|Du|2 =
|x|2
,
ρ2 − |x|2
1 + |Du|2 =
xi xj
;
− |x|2
2
ρ − |x|2 xi xj
xi xj
ij
= δij − 2 ;
g = δij −
2
2
2
ρ
ρ − |x|
ρ
gij = δij +
ρ2
ρ2
;
ρ2 − |x|2
MEAN CURVATURE FLOW
p
ν=
ρ2 − |x|2
ρ
39
p 2
ρ − |x|2
x
−p
,
.
,1 =
ρ
ρ
ρ2 − |x|2
−x
Thus
2
Dij
u
∂
∂
xi
u =
−p
=
∂xi
∂xj
ρ2 − |x|2
p
x
j
δij ρ2 − |x|2 − xi − √ 2 2
ρ −|x|
=−
=
ρ2 − |x|2
x i xj
δij
− 2
,
= −p
2
2
(ρ − |x|2 )3/2
ρ − |x|
∂
=
∂xj
therefore
p
ρ2 − |x|2
xi xj
δij
1
xi xj
p
+
hij = −
= − δij + 2
.
ρ
ρ
ρ − |x|2
ρ2 − |x|2 (ρ2 − |x|2 )3/2
Using that H = g ij hij , we get:
X
xi xj
xi xj 1
ij
δij + 2
H = g hij = −
δij − 2
=
2
ρ
ρ
ρ
−
|x|
i,j
P
2 2 xi x j
x i xj
1
i,j xi xj
− 2 δij − 2 2
=
= − δij δij + δij 2
ρ
ρ − |x|2
ρ
ρ (ρ − |x|2 )
|x|2
|x|4
|x|2
1 X
− 2 − 2 2
δij + 2
=
=−
ρ i,j
ρ − |x|2
ρ
ρ (ρ − |x|2 )
2
2
2
1
|x|4
2 ρ − ρ + |x|
= − n + |x| 2 2
− 2 2
=
ρ
ρ (ρ − |x|2 )
ρ (ρ − |x|2 )
n
=− ,
ρ
P
P
where we have used δij δij = i,j δij , δij xi xj = |x|2 , and i,j x2i x2j =
P 2P 2
2
2
4
i xi
j xj = |x| |x| = |x| .
Taking into account the previous computations the PDE (5.4) leads
to a partial differential equation that we
that we do not
qsolve(notice
2
2
ρ(t) − x(t) ) :
work now with u(x) but with u(x, t) =
p
∂u
n
q
= H 1 + |Du|2 = −
∂t
ρ(t)
ρ(t)
n
2 = − q
2 ,
2 2 ρ(t) − x(t)
ρ(t) − x(t)
40
FRANCISCO MARTIN AND JESUS PEREZ
in other words,
∂u
n
=−
,
∂t
u(x, t)
which is an ODE that we can integrate:
p
u2
= −nt + C ⇒ u(x, t) = K(x) − 2nt,
2
where K(x) is a “constant”
p (which depends on x but not on t). As the
initial data is u(x, 0) = ρ2 − |x|2 , then we deduce
q
u(x, t) =
ρ2 − 2nt − |x|2 , x ∈ B(0, ρ2 − 2nt).
udu = −ndt ⇒
Finally, notice that:
(5.8)
ρ2 − 2nt ≥ 0 ⇔
ρ2
≥ t,
2n
then, if the starting sphere is Sρ (0), the collapse occurs at time t =
(see Figure 2.)
ρ2
2n
Let us turn our attention to equation (5.7):
∂u
Di uDj u
2
= δij −
Dij
u.
∂t
1 + |Du|2
If we compare it with the definition of the operator (4.1) we obtain
that (in this particular case):
pi pj
(therefore it does not depend on z),
aij (X, z, p) = δij −
1 + |p|2
a(X, z, p) ≡ 0.
We claim that (5.7) defines a parabolic operator, that is, that the matrix whose coefficients are aij (X, z, p) is positive definite. Indeed,
• At p = 0, aij = δij , which is the identity matrix;
• If p =
6 0, consider x ∈ Rn \ {0}.
pi pj
xi p i p j xj
ij
xi a (X, z, p)xj = xi δij −
xj = xi δij xj −
=
2
1 + |p|
1 + |p|2
(xi pi )(pj xj )
hx, piRn hx, piRn
= hx, xiRn −
= hx, xiRn −
=
2
1 + |p|
1 + |p|2
hx, pi2
= |x|2 −
.
1 + |p|2
MEAN CURVATURE FLOW
41
Figure 2. A family of concentric spheres collapsing at
one point.
Taking this into account,
xi aij (X, z, p)xj > 0 ⇔ |x|2 −
hx, pi2
> 0 ⇔ hx, pi2 < |x|2 (1 + |p|2 ),
1 + |p|2
and this last inequality holds by the Cauchy-Schwarz inequality
(in Rn ): hx, pi2 ≤ hx, xihp, pi2 = |x|2 |p|2 , and because, obviously, |x|2 |p|2 < |x|2 (1 + |p|2 ) (recall that |x| =
6 0).
At this point we are ready to prove the comparison principle.
Theorem 5.3 (Comparison principle). Let M0 and N0 be compact,
embedded hypersurfaces, without boundary, in Rn+1 that do not intersect. If Mt and Nt are their respective evolutions by the mean curvature
flow, then they never intersect.
42
FRANCISCO MARTIN AND JESUS PEREZ
First proof. We proceed by contradiction. Assume Mt and Nt first intersect at time t0 at a point p (which is an interior point of both surfaces
because , by assumption, none of them have boundary.) Then both hypersurfaces have the same tangent plane at the point p, otherwise this
would not be your first point of contact. Then Mt0 y Nt0 can be expressed locally about p as graphs of functions, ut0 and vt0 respectively.
As both hypersurfaces have the same tangent plane at p, also have the
same unit normal vector at p, except perhaps by the sign. Considering
the same orientation on Mt0 and Nt0 we can compare them. Note that,
as t0 is the first point of contact, just a moment before both hypersurfaces do not intersect, , so either vt < ut or vt > ut . Without loss of
generality (since this depends on the choice of the Gauß map) we can
assume vt > ut . So, just a moment before t0 , there exists ε > 0 such
that vt − ut > ε.
Now we apply the mean curvature flow starting in that instant before t0 . As we have seen before, we know that ut and vt verifies the
quasilinear parabolic equation:
∂u
Di u, Dj u
2
Pu=−
+ δij −
Dij
u = 0.
∂t
1 + |Du|2
As we also check before, P can be seen as an operator satisfying the hypotheses of the comparison principle for parabolic operators (Theorem
4.3.) It is obvious that vt − is also a solution of the above equation.
Moreover, ut y vt − satisfy the assumptions of Theorem 4.3 in a neighborhood of the point p, then we deduce that ut ≤ vt − (< vt ) in this
particular neighbourhood of p, which contradicts u(p, t0 ) = v(p, t0 ).
This contradiction completes the proof.
As the reader can see, the idea in the proof of the above theorem is
very simple. However, we may get a better result if we work a little
more on the techniques to compare solutions of a partial differential
equation of parabolic type. For this purpose we are going to follow
Mantegazza’s monograph [Man11]. But first recall the concept of locally Lipschitz function, then we will use it in a previous result known
as Hamilton’s Trick.
Definition 5.4. Let (A, d) be a metric space. A function f : A → R is
defined to be Lipschitz (globally on A) if there exists a constant L > 0
such that:
|f (x) − f (y)| ≤ L · d(x, y) ∀x, y ∈ A.
MEAN CURVATURE FLOW
43
We will say that f is locally Lipschitz if for each x0 ∈ A there exist
U0 a neighborhood of x0 and a constant L0 > 0 such that:
|f (x) − f (y)| ≤ L0 · d(x, y) ∀x, y ∈ U0 .
Remark 5.5.
(1) If we want to be more precise, then we say that f is (locally)
L-Lipschitz, specifying the Lipschitz constant.
(2) It can be shown that a Lipschitz function f : R → R is differentiable almost everywhere (eg, proving that a Lipschitz function
is absolutely continuous). Later we will use this fact.
Lemma 5.6 (Hamilton’s Trick). Let u : M × (0, T ) → R a C 1 function such that for each t0 there exist δ > 0 and a compact subset
K ⊂ M \ ∂M such that for any t ∈ (t0 − δ, t0 + δ) the maximum
umax (t) := maxp∈M u(p, t) is reached at least at one point of K.
Then, the function umax is locally Lipschitz in (0, T ) and for each t0 ∈
(0, T ) where it is differentiable we have:
dumax (t0 )
∂u(p, t0 )
=
dt
∂t
where p ∈ M \ ∂M is any interior point where u(·, t0 ) reaches its maximum.
Proof. Consider t0 ∈ (0, T ). Let δ and K be as in the hypotheses of
the lemma.
We start by showing that u|K×(t0 −δ,t0 +δ) is Lipschitz with respect to t.
Fix p ∈ K, then we have to deduce the existence of a constant C > 0
such that if t1 < t2 in (t0 − δ, t0 + δ) then
|u(p, t2 ) − u(p, t1 )| ≤ C|t2 − t1 |.
This is essentially a consequence of the Mean Value Theorem for functions of class C 1 . Indeed, without loss of generality we can assume
that [t0 − δ, t0 + δ] ⊂ (0, T ). As u is C 1 , we can apply the Mean Value
Theorem to the function u(p, ·) : [t0 − δ, t0 + δ] → R. Furthermore
∂u
: K × [t0 − δ, t0 + δ] → R is bounded (K is compact). This im∂t
≤ C for all
plies the existence of a constant C > 0 such that ∂u(p,t)
∂t
(p, t) ∈ K × [t0 − δ, t0 + δ]. Therefore,
u(p, t2 ) − u(p, t1 ) ∂u(p, t3 ) |t2 − t1 | = |u(p, t2 ) − u(p, t1 )| = ∂t |t2 − t1 |
t2 − t1
≤ C|t2 − t1 |,
44
FRANCISCO MARTIN AND JESUS PEREZ
as we wanted to prove.
Let us see that umax is locally Lipschitz in (0, T ). Take t0 in (0, T )
joint with δ and K provided by the hypotheses of the lemma. Consider
0 < < δ. Taking into account that u|K×(t0 −δ,t0 +δ) is Lipschitz with
respect to t, we have
umax (t0 + ) = u(q, t0 + ) ≤ u(q, t0 ) + C ≤ umax (t0 ) + C,
for some q ∈ K (this point q exists by hypothesis). So
umax (t0 + ) − umax (t0 )
≤ C.
Analogously,
umax (t0 ) = u(p, t0 ) ≤ u(p, t0 + ) + C ≤ umax (t0 + ) + C,
for a certain p ∈ K. Therefore
umax (t0 ) − umax (t0 + )
≤ C.
Summarizing, we have showed that 0 < < δ,
|umax (t0 ) − umax (t0 + )| ≤ C|(t0 + ) − t0 |.
If we consider −δ < < 0, then we can prove in a similar way that:
|umax (t0 ) − umax (t0 + )| ≤ C|(t0 + ) − t0 |.
Thus, we have got that umax is locally Lipschitz in (0, T ), which implies
that it is differentiable a.e. in t ∈ (0, T ).
Finally, take a point t0 ∈ (0, T ) where umax is differentiable. From
our assumptions, there exists p ∈ M \ ∂M so that umax (t0 ) = u(p, t0 ).
By the Mean Value Theorem, for each 0 < < δ there exists ξ ∈
(t0 , t0 + ) such that u(p, t0 + ) = u(p, t0 ) + ∂u(p,ξ)
. Therefore
∂t
umax (t0 + ) ≥ u(p, t0 + ) = u(p, t0 ) + ∂u(p, ξ)
∂u(p, ξ)
= umax (t0 ) + ,
∂t
∂t
from which, as > 0, we can deduce
umax (t0 + ) − umax (t0 )
∂u(p, ξ)
≥
.
∂t
Taking limit, as → 0 we get
∂u(p, ξ)
u0max (t0 ) ≥
.
∂t
Applying just the same argument for −δ < < 0 we conclude
umax (t0 + ) − umax (t0 )
∂u(p, ξ)
≤
,
∂t
MEAN CURVATURE FLOW
45
and taking limit again, as → 0 what we get now is that:
∂u(p, ξ)
u0max (t0 ) ≤
.
∂t
Summarizing, u0max (t0 ) =
∂u(p,ξ)
.
∂t
Corollary 5.7. Hamilton’s trick also holds if we consider umin (t) :=
minp∈M u(p, t) instead of umax .
Proof. We just consider v := −u.
Theorem 5.8 (Comparison principle). Let ϕ : M1 × [0, T ) → Rn+1
and ψ : M2 × [0, T ) → Rn+1 be two hypersurfaces moving by mean
curvature. Suppose that M1 is compact, M2 is complete and that ψt is
proper11 for any t ∈ [0, T ).Then the distance between the hypersurfaces
is non-decreasing in time.
Proof. First notice that, as ϕt (M1 ) is compact and ψt (M2 ) is properly
immersed, then the distance between both hypersurfaces at time t is
given by:
d(t) = min |ϕ(p, t) − ψ(q, t)|.
p∈M1 ,q∈M2
We want to apply Lemma 5.6, but the problem is that the Euclidean
norm has differentiability problems at the origin. So we are going to
consider M := M1 × M2 and the function u : M × (0, T ) → R given
by u(p, q, t) := |ϕ(p, t) − ψ(q, t)|2 = hϕ(p, t) − ψ(q, t), ϕ(p, t) − ψ(q, t)i
Notice that
umin (t) := min |ϕ(p, t) − ψ(q, t)|2 = d(t))2 .
(p,q)∈M
So, it suffices to prove that umin (t) is a non-decreasing function.
Step 1. The function u satisfies the hypotheses of Corollary 5.7 (which
are the same as Lemma 5.6.)
Clearly u is C 1 (even more, it is smooth.)
Fix t0 , we want to obtain the existence of δ > 0 and a compact
subset K ⊂ M \ ∂M such that for any t ∈ (t0 − δ, t0 + δ) the minimum
umin (t) = min(p,q)∈M u(p, q, t) is attained at one point of K.
11
ψt :
ψt−1 (K)
M2 → Rn+1 is defined proper if for any compact subset K ⊂ Rn+1 then
is also compact.
46
FRANCISCO MARTIN AND JESUS PEREZ
Take δ > 0 satisfying [t0 − δ, t0 + δ] ⊂ (0, T ). Notice that umin ([t0 −
δ, t0 + δ]) is bounded, because M1 is compact. So, there is a positive
constant β > 0 such that
umin ([t0 − δ, t0 + δ]) ⊂ [0, β].
Let us define:
K := u−1 [0, β]) ∩ (M1 × M2 × [t0 − δ, t0 + δ]) .
We have to prove that K is compact. Clearly, K is closed. If we
prove that K is bounded, then we have done. Assume that K is not
bounded. As M1 and [t0 − δ, t0 + δ] are compact, this means that π2 (K)
is unbounded, where π2 is the second canonical projection. Thus, we
take {qn } a sequence in π2 (K) such that neither the sequence itself nor
any subsequence is bounded. Consider pn ∈ M1 and tn ∈ [t0 − δ, t0 + δ]
such that (pn , qn , tn ) ∈ K. Up to taking a subsequence, we can assume
that {tn } → t0 ∈ [t0 − δ, t0 + δ]. From the definiton of K, we have that
ψtn (qn ) belongs to the set
n
p o
n+1
K := x ∈ R
: distRn+1 (x, ϕ (M1 × [t0 − δ, t0 + δ])) ≤ β ,
which is compact. As we are assuming that ψt0 is proper, then K0 =
0
ψt−1
0 (K) is also compact. So, any limit point of {qn } must lie on K , which
is absurd because we are assuming that this sequence is unbounded.
This contradiction proves that K is compact.
Hence, Corollary 5.7 gives us the function umin is locally Lipschitz in
(0, T ) and for each t0 ∈ (0, T ) where it is differentiable we have:
d
∂
umin (t0 ) = u(p0 , q0 , t0 ),
dt
∂t
where (p0 , q0 ) ∈ M1 ×M2 is any point where u(·, t) reaches its minimum.
Step 2. Let (p0 , q0 ) be a minimum of u(·, t), then
∂
u(p0 , q0 , t0 ) ≥ 0.
∂t
We distinguish two cases:
∂
Case 1. u(p0 , q0 , t0 ) = 0, then ∂t
u(p0 , q0 , t0 ) cannot be negative,
otherwise u(p0 , q0 , t) would be negative for t in a small interval [t0 , t0 +
s), which is absurd.
Case 2. u(p0 , q0 , t0 ) 6= 0. As (p0 , q0 ) is a minimum of u, we have
that ϕ(p0 , t0 ) − ψ(q0 , t0 ) is normal to both hypersurfaces; ϕt0 (M1 ) and
ψt0 (M2 ). In other words, the respective tangent hyperplanes are parallel.
MEAN CURVATURE FLOW
47
This allows us to write the respective hypersurfaces (locally around
p0 and q0 ) as graphs of functions, f and h, defined on (a part of) one of
their tangent spaces, Π, in a small time interval (t0 − , t0 + ). Without
loss of generality, we can fix a reference in Rn+1 so that {e1 , . . . , en },
the canonical basis of Rn , is a base of the hyperplane Π. Assume that
ϕ(p0 , t0 ) = (0, f (0, t0 )) and ψ(q0 , t0 )) = (0, h(0, t0 )) with f (0, t0 ) >
h(0, t0 ). Note that in this reference we have
en+1 =
ϕ(p0 , t0 ) − ψ(q0 , t0 )
.
|ϕ(p0 , t0 ) − ψ(q0 , t0 ))|
From (5.7) we have:
(5.9)
∂f
∇2 f (Df, Df )
= ∆f −
,
∂t
1 + |Df |2
∂h
∇2 h(Dh, Dh)
= ∆h −
.
∂t
1 + |Dh|2
On the other hand, as the function f (x, t0 ) − h(x, t0 ) has a minimum
at x = 0, then its gradient vanishes at this point and the Hessian is
positive semidefinite at x = 0. In particular, the Laplacian of f (x, t0 )−
h(x, t0 ) is non-negative.
0 = ∇ f (0, t0 ) − h(0, t0 ) = ∇f (0, t0 ) − ∇h(0, t0 ),
0 ≤ ∆ f (0, t0 ) − h(0, t0 ) = ∆f (0, t0 ) − ∆h(0, t0 ).
Moreover, from our choice of the set of coordinates, we also have;
∇f (0, t0 ) = ∇h(0, t0 ) = 0.
Using (5.1), and Df (0, t0 ) = ∇f (0, t0 ) = 0, we have
Df
H(0, t0 ) = div
(0, t0 ) =
1 + |Df |2
2
Di f Dij
f Dj f
∆f
= p
−
(0, t0 ) = ∆f (0, t0 ).
1 + |Df |2 (1 + |Df |2 )3/2
Again, all we got for f is also valid for h.
If we denote as ν ϕ and ν ψ the unit normal fields associated to the ϕ
and ψ, respectively, we can write
H ϕ (p0 , t0 )ν ϕ (p0 , t0 ) = 0, ∆f (0, t0 ) ,
H ψ (q0 , t0 )ν ψ (q0 , t0 ) = 0, ∆h(0, t0 ) ,
If we multiply by en+1 , then we obtain the following equalities:
∆f (0, t0 ) = H ϕ (p0 , t0 )hν ϕ (p0 , t0 ), en+1 i,
∆h(0, t0 ) = H ψ (q0 , t0 )hν ψ (q0 , t0 ), en+1 i.
48
FRANCISCO MARTIN AND JESUS PEREZ
Hence,
(5.10) hH ϕ (p0 , t0 )ν ϕ (p0 , t0 ) − H ψ (q0 , t0 )ν ψ (q0 , t0 ), en+1 i =
= ∆f (0, t0 ) − ∆h(0, t0 ) ≥ 0.
∂
Now, we can compute ∂t
u(p0 , q0 , t0 ), taking into account that ϕ and
ψ are solutions of the MCF equation:
∂
∂
u(p0 , q0 , t) = hϕ(p0 , t) − ψ(q0 , t), ϕ(p0 , t) − ψ(q0 , t)i =
∂t ∂t
∂ϕ(p0 , t) ∂ψ(q0 , t)
=2
−
, ϕ(p0 , t) − ψ(q0 , t) =
∂t
∂t
= 2hH ϕ (p0 , t)ν ϕ (p0 , t) − H ψ (q0 , t)ν ψ (q0 , t), ϕ(p0 , t) − ψ(q0 , t)i =
= 2 H ϕ (p0 , t)ν ϕ (p0 , t) − H ψ (q0 , t)ν ψ (q0 , t), en+1 |ϕ(p0 , t0 ) − ψ(q0 , t0 )|,
ϕ(p0 ,t0 )−ψ(q0 ,t0 )
where we have used that en+1 = |ϕ(p
.
0 ,t0 )−ψ(q0 ,t0 ))|
Evaluating the last equality at t = t0 and taking (5.10) into account,
∂
we obtain that ∂t
u(p0 , q0 , t0 ) ≥ 0.
As we noted at the beginning of the proof, the second step proves
that d(t) is non-decreasing, which completes the demonstration.
Remark 5.9 ([MSHS14]). We would like to point out that the properness assumption cannot be relaxed with that of completeness. Indeed,
take as f : M1 → R3 be the unit euclidean sphere and as g : M2 → R3
a complete minimal surface lying inside the unit ball. Such examples
were first constructed by Nadirashvili [Nad96]. Obviously f and g do
not have intersection points. However, under the mean curvature flow,
f shrinks to a point in finite time while g remains stationary.
The following result is an immediate consequence of the previous theorem, but it is useful to state it independently. It has a very geometric
meaning and it will be used in several arguments.
Corollary 5.10 (Inclusion principle). Let ϕ : M1 × [0, T ) → Rn+1
and ψ : M2 ×[0, T ) → Rn+1 two compact hypersurfaces moving by mean
curvature. Assume that the inner domain of ϕ(M1 , 0) strictly contains
ψ(M2 , 0). Then, ψ(M2 , t) stays strictly inside of ϕ(M1 , t) for any t ∈
[0, T ).
Among other things, the above corollary has an interesting consequence, which has a decisive influence on the study of mean curvature
MEAN CURVATURE FLOW
49
flow; the existence of singularities in finite time for the flow of a compact hypersurface.
Corollary 5.11 (Existence of singularities in finite time). Let M
be a compact hypersurface in Rn+1 . If Mt represents its evolution by
the mean curvature flow, then Mt must develop singularities in finite
time. Moreover, if we denote this maximal time as Tmax , then we have
that:
2 n Tmax ≤ (diamRn+1 (M ))2 .
Proof. As M in compact, then it can be included inside an open ball
B(p, ρ). So, M must develop a singularity before the flow of Snp collapses
at the point p, otherwise we would contradict Corollary 5.10. The upper
bound of Tmax is just a consequence of (5.8).
A natural question is: What can we say when M is not compact?
In this case, we can have long time existence. A trivial example is
the case of a complete, properly embedded minimal hypersurface M
in Rn+1 . Under the mean curvature flow, M remains stationary, so
the flow exists for any value of t. If we are looking for non-stationary
examples, then we can consider the following example:
Example 5.12 (Grim hyperplanes). Consider the euclidean product M = Γ × Rn−1 , where Γ is the grim reaper in R2 represented by
the immersion f : (−π/2, π/2) → R2 given by
f (x) = (x, 1 − log cos x).
If we move M by mean curvature we get Mt = φt (M ) + t · en+1 , where
again {e1 , . . . , en+1 } represents the canonical basis of Rn+1 and φt :
M → M is a (tangent) diffeomorphism. In other words, M moves by
vertical translations, that do not have singularities for any value of t.
By definition, we say that M is a translating soliton in the direction
of en+1 . More generally, any translator in the direction of en+1 which
is a Riemannian product of a planar curve and an euclidean space
Rn−1 can be obtained from this example by a suitable combination of
a rotation and a dilation (see [MSHS14] for further details.) Each of
these translators will be called a grim hyperplane.
The ideas introduced in the proof of Theorem 5.8 can be used to get
a very interesting result.
Theorem 5.13 (Embeddedness principle). Let F : M × [0, T ) → Rn+1
a MCF, with M compact. If F0 : M → Rn+1 is an embedding, then Ft
is an embedding for any t ∈ [0, T ).
50
FRANCISCO MARTIN AND JESUS PEREZ
Figure 3. A grim hyperplane
Proof. Let us define
A := {t ∈ [0, T ) : Fs is an embedding for all s ∈ [0, t]}.
Following a classical connectedness argument, we are going to prove
that A is open, closed and non-empty, so A = [0, T ).
Notice that A is not empty, since 0 ∈ A .
Moreover A is open. Indeed, take t ∈ A and assume his set is not
open. That would mean the existence of a sequence {tj }j∈N & t and
sequences of points {pj }j∈N and {qj }j∈N in M such that F (pj , tj ) =
F (qj , tj ), pj 6= qj , for all j ∈ N. As M is compact, then (up to a
subsequence) we can assume that {pj }j∈N → p and {qj }j∈N → q. We
have two possibilities; either p = q or p 6= q.
If p 6= q, then we would conclude that Ft is not an embedding which
id contrary to t ∈ A .
If p = q, we can take U a neighborhood of p in M such that Ftj |U is
an embedding for all j ∈ N. This is possible since Ftj are immersions
and they converge to Ft which is an embedding. If j is large enough,
then we have that pj and qj belong to U , which is absurd because
F (pj , tj ) = F (qj , tj ). This contradiction proves the openness of A .
MEAN CURVATURE FLOW
51
Note that so far we have not used that F : M × [0, T ) → Rn+1 moves
by its mean curvature.
Finally, we are going to prove that A is closed, or equivalently that
sup A = T. We proceed again by contradiction. Suppose t0 = sup A <
T.
So, we consider W ⊂ M × M ,
W := {(p, q) ∈ M × M : F (p, t0 ) = F (q, t0 ), p 6= q}.
Then, W is a closed subset disjoint from the diagonal ∆ = {(p, p) :
p ∈ M }. Indeed, if (p, q) is a point in W , we only have to check that
p 6= q to guarantee that (p, q) ∈ W (the other condition is closed.) Let
{(p, qn )} ⊂ W such that {(p, qn )} → (p, q). If p = q, then take a chart
(V, φ) around p in M . There exists n0 ∈ N such that, for any n ≥ n0 ,
one has pn , qn ∈ V . Up to a subsequence, we have that
φ(pn ) − φ(qn )
vn :=
|φ(pn ) − φ(qn )|
converges to some unitary vector v ∈ Sn .. Then
d Ft0 ◦ φ−1
F (pn , t0 ) − F (qn , t0 )
= 0,
n→∞
|φ(pn ) − φ(qn )|
(v) = lim
φ(p)
which is contrary to the assumption that Ft0 is an immersion. So p 6= q
and W is closed and, from its own definition, W ∩ ∆ = ∅.
Hence, we can take a regular domain Ω ⊂ M × M such that
W ⊂ Ω ⊂ Ω ⊂ M × M \ ∆.
Let us define u : Ω × (0, t0 ) −→ R
u(p, q, t) := |F (p, t) − F (q, t)|2 .
As u| ∂Ω × (0, t0 ) > σ > 0, for some12 σ, then we can guarantee the
existence of t1 ∈ (0, t0 ) such that u : Ω × (t1 , t0 ) −→ R satisfies the
hypotheses of Hamilton’s trick (Lemma 5.6). So, we have that
umin : (t1 , t0 ) −→ R
is locally Lipschitz and
d
∂
umin (t) =
u(p0 , q0 , t),
dt
∂t
12recall
that ∂Ω is compact
t ∈ (t1 , t0 ),
52
FRANCISCO MARTIN AND JESUS PEREZ
where (p0 , q0 ) is any point where u(·, t) reaches its minimum. But
reasoning as in the proof of Theorem 5.8, we deduce that
∂
u(p0 , q0 , t) ≥ 0, t ∈ (t1 , t0 ).
∂t
This is absurd because umin (t0 ) = 0 and umin (t) > 0, for any t ∈ (t1 , t0 ).
This contradiction proves that t0 = T and concludes the proof.
6. Area Estimates and Monotonicity Formulas
Mean curvature flow can be also formulated in an integral way. Using
this formulation, we will deduce area estimates and we focus our attention on monotonicity formulas and its consequences. These formulas
are considered the most important tools in the study of the formation
and structure of singularities.
Throughout all this section we denote by (Mt )t∈I , where I is an open
interval of R (usually I = (0, T )), a family of smooth hypersurfaces
moving, properly embedded by the mean curvature. Recall that this
just means that there is a family of smooth, proper, embeddings13
F : M n × I → Rn+1 , where:
∂
~
F (p, t) = H(p,
t) ∀(p, t) ∈ M n × I,
∂t
As we already did in the previous section, we denote Mt := F (M n , t).
(6.1)
To derive an integral version of the mean curvature flow is essential
to recall how they change the area elements of the hypersurfaces Mt .
These are given by
p
dµt = det gij dµM
where dµM denotes the area form of M . This volume form determines
a measure on Mt which essentially coincides with the n- dimensional
Hausdorff measure of Rn+1 , H n , restricted to the hypersurface Mt .
For details more details about integration on Riemannian manifold we
recommend [Cha93, Mor00]. So, given a measurable function f : Mt →
R, we denote:
Z
Z
Z
n
f≡
f (x)dH (x) =
f (F (p, t))dµt .
Mt
13Theorem
Mt
M
5.13 precisely says that if the initial hypersurface is compact and
embedded, then they remain embedded along the flow.
MEAN CURVATURE FLOW
53
The evolution of the area element under the mean curvature flow was
already computed in (3.2), when we were considering the evolution of
the intrinsic geometry
(6.2)
∂
~ 2 dµt
dµt = −|H|
∂t
for all t ∈ I.
If we consider a domain Ω ⊂ Rn+1 and f ∈ C01 (Ω) (i.e., f is a C 1
function with compact support), then
∂
∂
∂
(f (F (p, t))dµt ) = hD f, F (p, t)idµt + f (F (p, t)) dµt =
∂t
∂t
∂t
using that F is a solution of (6.1) and (6.2), we deduce
~
~ 2
= hD f, Hidµ
t − f (F (p, t))|H| dµt .
In this way we have proven the following:
Theorem 6.1 (Integral form for the Mean Curvature Flow,
[Bra78]). Given (Mt )t∈I a mean curvature flow contained in Ω ⊂ Rn+1
we have
Z Z
d
~ − |H|
~ 2f
hD f, Hi
f=
(6.3)
dt Mt
Mt
for any t ∈ I and f ∈ C01 (Ω).
If M is compact, we can obtain the following corollary
Corollary 6.2 (Area decreasing property). The mean curvature
flow of a compact hypersurface decreases the area. To be more precise,
given a family (Mt )t∈I of compact solutions of the mean cuvature flow
we have that:
Z
d
~ 2
Area(Mt ) = −
|H|
dt
Mt
for all t ∈ I.
Remark 6.3 (Brakke’s solutions). Theorem 6.1 can be used as a
definition of mean curvature flow, because any family of smooth, properly embedded, hypersurfaces verifying (6.3) also is a solution in the
sense of (6.1) (which is the one we have been considering throughout
these notes). In fact, (6.3) is the motivation of the concept of “Brakke’s
54
FRANCISCO MARTIN AND JESUS PEREZ
solution”. Brakke defined the mean curvature flow for generalized hypersurfaces (called integral varifolds in the language of Geometric Measure Theory.)
Following the ideas of K. Ecker [Eck04], we will derive all the results
of this section by substituting appropriate test functions in identity
(6.3).
Definition 6.4 (Time-dependent test function). Let Ω be an open
set in Rn+1 , I an interval in R and φ ∈ C 1 (Ω × I). We say that φ
is a time-dependent test function if it satisfies φ(·, t) ∈ C02 (Ω) and
∂
φ(·, t) ∈ C00 (Ω) for all t ∈ I.
∂t
Notice that if φ : Ω → R is like in the previous definition, then φ and
are integrable on Mt . Moreover, using the divergence theorem14 we
have that , for any t ∈ I,
Z
Z
~ Dφi = 0.
divMt Dφ + hH,
(6.4)
∆Mt φ =
∂φ
∂t
Mt
Mt
For the sake of simplicity, in what follows we will write
R
of Mt φ(·, t).
R
Mt
φ instead
For a test function we have that:
(6.5)
d
[φ(F (p, t), t) · dµt ] =
dt ∂φ
~ 2 · dµt
~
(F (p, t), t) · dµt − φ(F (p, t), t) · |H|
hDφ, Hi +
∂t
|
{z
}
= dφ
dt
Hence, combining (6.1), (6.4) and (6.5) we immediately obtain the following result.
Proposition 6.5 ([Bra78, Eck04]). Let (Mt )t∈I be a mean curvature
flow in Ω ⊂ Rn+1 and let φ be a time-dependent test function on Ω × I.
14Here
we use that:
~ Df i,
∆Mt f = divMt Df + hH,
where f : Ω × I → R is sufficiently regular. D denotes the gradient in Rn+1 .
MEAN CURVATURE FLOW
55
Then, the following equalities hold:
Z
Z
Z
∂φ
d
dφ
2
~ Dφi − |H|2 φ,
(6.6)
− |H| φ =
+ hH,
φ=
dt Mt
dt
∂t
Mt
Mt
d
(6.7)
dt
Z
φ=
Mt
Z
Mt
∂φ
− divMt Dφ − |H|2 φ =
∂t
Z
Mt
d
− ∆Mt φ − |H|2 φ
dt
and
d
dt
(6.8)
Z
Z
φ=
Mt
Mt
d
+ ∆Mt φ − |H|2 φ.
dt
As an immediate consequence of this proposition we conclude that in
a certain sense (specified with an additional hypothesis in the following
result) the area of the solutions of mean curvature flow also decreases
locally.
Corollary 6.6 (Local area decay, [Bra78, Eck04]). If the test function φ verifies
d
∂φ
(6.9)
− ∆Mt φ =
− divMt Dφ ≤ 0,
dt
∂t
then we have:
d
dt
(6.10)
Z
Z
φ≤−
Mt
~ 2φ
|H|
Mt
for each t ∈ I.
Definition 6.7. Spherically shrinking test functions. An important
class of test functions is the following: given ρ > 0 and (x, t) ∈ Rn+1 ×I
we consider15
3
|x|2 + 2nt
ϕρ (x, t) = 1 −
ρ2
+
and its translations:
ϕ(x0 ,t0 ),ρ (x, t) = ϕρ (x − x0 , t − t0 ).
Lemma 6.8. The function ϕ(x0 ,t0 ),ρ satisfies:
15As
usual f+ means f+ (x) := max(f (x), 0).
56
FRANCISCO MARTIN AND JESUS PEREZ
Figure 4. The evolution of ϕρ (·, t), as t → 0.
p
(i) spt(ϕ(x0 ,t0 ),ρ (·, t)) ⊂ B(x0 , ρ2 − 2n(t0 − t)).
(ii) The maximum of ϕ(x0 ,t0 ),ρ (·, t) is reached at x = x0 , where
ϕ(x0 ,t0 ),ρ (x0 , t) =
ρ2 − 2n(t − t0 )
.
ρ2
(iii) Moreover, the following inequality holds:
(6.11)
d
− ∆Mt ϕ(x0 ,t0 ),ρ ≤ 0.
dt
Proof. The first two properties are trivial. Regarding item (iii), we are
going to write ϕ(x0 ,t0 ),ρ = g ◦ u, where g : R −→ R is given by
3
r
,
g(r) = 1 − 2
ρ +
and u : Rn+1 × (−∞, t0 ) −→ R
u(x, t) = |x − x0 |2 + 2n(t − t0 ).
MEAN CURVATURE FLOW
57
~ Dui. It is clear that Du =
If we compute ∆Mt u = divMt Du + hH,
2(x − x0 ), so
~
∆Mt u = 2 n + hH, x − x0 i .
On the other hand, taking into account that Mt is solution of the MCF
equation, we have
d
~ x − x0 i + 2n.
u = 2hH,
dt
From the last two equalities, we get
d
(6.12)
− ∆Mt u = 0.
dt
Taking into account that
∆Mt (g(u)) = g 0 (u) · ∆Mt u + g 00 (u) · |∇u|2 ,
and
d
d
g(u) = g 0 (u) · u.
dt
dt
Therefore, using (6.12), we get
d
− ∆Mt g(u) =
dt
d
0
g (u)
− ∆Mt u − g 00 (u) · |∇u|2 = −g 00 (u) · |∇u|2 .
dt
Finally, notice that
6
g (r) = 4
ρ
00
r
≥ 0,
1− 2
ρ +
which concludes the proof.
Using these test functions, the following local estimate of the area is
obtained:
Proposition 6.9 (Local area estimate, [Bra78, Eck95]). Let Mt be
a smooth, properly embedded, solution of the mean curvature flow in
ρ2
, t0 ). Then we have
B(x0 , ρ) × (t0 − 8n
Z
Z t
ρ ~ 2
Area Mt ∩ B(x0 , ) +
|H|
ρ2
2
t0 − 8n
Mt ∩B(x0 , ρ2 )
≤ 8 Area Mt − ρ2 ∩ B(x0 , ρ)
0
8n
58
FRANCISCO MARTIN AND JESUS PEREZ
for all t ∈ [t0 −
ρ2
, t ].
8n 0
Proof. Take φ = ϕ(x0 ,t0 −ρ2 /(8n)),ρ . So, we integrate over the interval
ρ2
, t) the inequality given by Corollary 6.6 and obtain
(t0 − 8n
Z
Z t
Z
Z
~ 2 φ,
φ−
φ≤−
|H|
2
Mt
M
or equivalently,
Z
Z
φ+
Mt
ρ
t0 − 8n
ρ2
t0 − 8n
t
Z
2
ρ
t0 − 8n
Mt ∩B(x0 , ρ2 )
Mt ∩B(x0 , ρ2 )
~ 2 φ, ≤
|H|
Z
φ
M
ρ2
t0 − 8n
Now, note that φ(x, t) ≥ 1/8, whenever |x − x0 | ≤ ρ/2 and t ≤
t0 + ρ2 /(8n). So, we use this lower bound on the left-hand side of
the inequality. On the right-hand side, we apply that φ ≤ 1. This
completes the proof.
We would like to point out that, using Corollary 6.6, we can deduce
a “comparison principle” with the sphere.
Proposition 6.10. If we have a mean curvature flow Mt , t ∈ [0, T ),
satisfying
Area(M0 ∩ B(0, ρ)) = 0,
then
p
ρ2
Area(Mt ∩ B(0, ρ2 − 2nt)) = 0, for all t <
.
2n
Proof. By definition the function ϕρ is non-negative. It also follows immediately from its definition
that its support
p
p is the ball with center at
2
the origin and radius ρ − 2nt, i.e. B(0, ρ2 − 2nt). Then,
p the integral of ϕρ on Mt is the same as integrating it over Mt ∩B(0, ρ2 − 2nt)
. Also as ϕρ is non-negative, so its integralRis also non-negative. Then,
in this case, Corollary 6.6 says to us that Mt ϕρ is not decreasing in t
(and it is non-negative). Therefore, we conclude that if at the initial
time this function is zero, it must remain zero.
Remark 6.11. If Mt is compact, then we use ϕ(x, t) = (−ϕρ (x, t))3+
to deduce
p
Area(M0 \ B(0, ρ)) = 0 ⇒ Area(Mt \ B(0, ρ2 − 2nt)) = 0,
for t <
ρ2
.
2n
MEAN CURVATURE FLOW
59
One can check that the area estimate provided by Proposition 6.9 can
not be used to bound the ratio
Area(Mt ∩ B(x0 , ρ))
ρn
independently of ρ in a time interval of length proportional to ρ2 ; to
establish such an estimate valid in a longer time interval an more accurate result is needed (see [Eck04, p. 75].) In order to do this we need
the following definition:
Definition 6.12 (Backward Heat Kernels). Given x ∈ Rn+1 y
t < 0,
|x|2
1
4t
Φ(x, t) =
n e
(−4πt) 2
and its translations
|x−x0 |2
1
− 4(t −t)
0
Φ(x0 ,t0 ) (x, t) = Φ(x − x0 , t − t0 ) =
e
n
4π(t0 − t) 2
where x0 ∈ Rn+1 and t < t0 .
1
Ψ(x, t), where Ψ is the stanRemark 6.13. Note that Φ(x, t) = √−4πt
dard heat kernel in Euclidean space. Then, we have that:
∂
1
+ ∆Rn+1 Φ(x0 ,t0 ) =
Φ(x ,t ) .
(6.13)
∂t
2 (t − t0 ) 0 0
Bearing in mind the similarities of (2.2) with the standard heat equation, it makes sense that the function Φ can be useful to study the
mean curvature flow.
Lemma 6.14. In the previous setting, the following equality holds:
∂Φ
|∇⊥ Φ|2
+ divMt DΦ +
= 0.
∂t
Φ
Proof. Recall that divMt DΦ = ∆Rn+1 Φ − D2 Φ(ν, ν), where D2 Φ is the
Euclidean Hessian of Φ and ν is the Gauß map. On the other hand,
from its definition we trivially deduce that, in general,
Φ(x0 ,t0 )
· (x − x0 ),
DΦ(x0 ,t0 ) =
2(t − t0 )
and so
Φ(x0 ,t0 )
(6.14)
∇⊥ Φ(x0 ,t0 ) =
· (x − x0 )⊥ ,
2(t − t0 )
Φ(x0 ,t0 )
(6.15)
|∇⊥ Φ(x0 ,t0 ) | =
· hx − x0 , νi.
2(t − t0 )
60
FRANCISCO MARTIN AND JESUS PEREZ
So, using 6.13 and (6.15) (for t0 = 0 and x0 = 0), we deduce that
∂Φ
|∇⊥ Φ|2
+ divMt DΦ +
=
∂t
Φ
∂Φ
hx, νi2
+ ∆Rn+1 Φ − D2 Φ(ν, ν) +
Φ=
∂t
4t2
hx, νi2
1
Φ − D2 Φ(ν, ν) +
Φ.
2t
4t2
If we compute the Hessian term, we get:
1
1
D2 Φ(ν, ν) = 2 hx, νi2 Φ + hν, νiΦ.
4t
2t
Using that ν is unitary and substituting in (6.16), we conclude the
proof.
(6.16)
The next result is a monotonicity formula proved by Huisken in
[Hui90]. It deals with the monotonic behaviour of the integral over
Mt of the backward heat kernel Φ(x0 ,t0 ) (x, t).
Theorem 6.15 (Monotonicity formula). Consider (x0 , t0 ) ∈ Rn+1 ×
RR and (Mt )t∈I a smooth solution of the mean curvature flow satisfying
Φ
< ∞ ∀t ∈ I with t < t0 . Then, for t < t0 , we have
Mt (x0 ,t0 )
Z
Z ∇⊥ Φ(x0 ,t0 ) 2
d
~
Φ(x0 ,t0 ) = −
H − Φ(x ,t ) Φ(x0 ,t0 ) .
dt Mt
Mt
0 0
R
In particular, Mt Φ(x0 ,t0 ) is decrasing for any t < t0 .
Moreover, for x ∈ Mt and t < t0 the following equality holds:
∇⊥ Φ(x0 ,t0 ) (x, t)
(x − x0 )⊥
~
~
H(x) −
= H(x)
−
.
Φ(x0 ,t0 ) (x, t)
2(t − t0 )
Proof. From now on, and for simplicity, we will write Φ0 instead of
Φ(x0 ,t0 ) .
So, with this notation, we have:
d
∂Φ0
~ DΦ0 i + divMt DΦ0 + hH,
~ DΦ0 i =
+ ∆Mt Φ0 =
+ hH,
dt
∂t
(recall that, using Gauß equation, we have divMt DΦ0 = ∆Mt Φ0 −
~ DΦ0 i)
hH,
∂Φ0
~ DΦ0 i.
+ divMt DΦ0 + 2 hH,
=
∂t
MEAN CURVATURE FLOW
61
~ DΦ0 i = hH,
~ ∇⊥ Φ0 i and so,
Now we use that hH,
∂Φ0
d
~ ∇⊥ Φ0 i−
+ ∆Mt Φ0 =
+ divMt DΦ0 + 2 hH,
dt
∂t
2
2
⊥
⊥
∇
Φ
∇
Φ
0
0
~ −
~ −
H
Φ0 + H
Φ0 =
Φ0 Φ0 2
|∇⊥ Φ0 |2 ~
∇⊥ Φ0 ∂Φ0
~ 2 Φ0 .
+ divMt DΦ0 +
− H −
Φ0 + |H|
=
∂t
Φ0
Φ0 Using Lemma 6.14, we substitute to obtain that
2
∇⊥ Φ0 d
~ 2 Φ0 .
~
+ ∆Mt Φ0 = − H −
Φ0 + |H|
(6.17)
dt
Φ0 If Mt is compact, we use (6.8) to finish.
In the general case, we are going to use a test function φ, whose
properties will be determined later.
Z
Z d
dΦ0
dφ
2
~
(6.18)
φ · Φ0 =
Φ0 +
φ − |H| · φ · Φ0
dt
dt
dt
Mt
Mt
Now, we use that
divMt (φ∇Φ0 − Φ0 ∇φ) = φ∆Mt Φ0 − Φ0 ∆Mt φ.
Integrating over Mt , and using that φ has compact support, we obtain
(applying the divergence theorem)
Z
φ∆Mt Φ0 − Φ0 ∆Mt φ = 0.
(6.19)
Mt
Adding (6.18) + (6.19) we get
Z
d
φ · Φ0 =
dt
Mt
Z
d
d
2
~ Φ0
Φ0
− ∆Mt φ + φ
+ ∆Mt Φ0 − |H|
dt
dt
Mt
Substituting (6.17) in the previous equality, it becomes:
Z
d
(6.20)
φ · Φ0 =
dt
Mt
2
Z
d
∇⊥ Φ0 ~
− ∆Mt φ − H −
Φ0 φ.
Φ0
dt
Φ0 Mt
At this point, we are going to make our choice of φ. Fix R > 0, then
we take φ a smooth function satisfying:
62
FRANCISCO MARTIN AND JESUS PEREZ
• χB(x0 ,R) ≤ φ ≤ χB(x0 ,2 R) .
• R|Dφ| + R2 |D2 φ| ≤ C0 , where C0 is a positive constant.
For the existence of such a function see [Ilm95]. Hence, for this particular φ, we have:
d
≤ κ ,
φ
−
∆
M
t
dt
R2
for κ a constant depending on n and C0 . At this point, we can apply
the theorems of convergence under the integral sign and take limit, as
R → +∞. The first integral of the right-hand side goes to zero. Taking
into account that limR→∞ φ ≡ 1, we conclude the proof.
Roughly speaking, the monotonicity formula means that the area of
a hypersurface which moves by the mean curvature near any point is
non-increasing on any scale. Actually, it strictly decreases unless the
hypersurface is homothetically shrinking around the point x0 .
Remark 6.16. Notice that the assumption
Z
Φ(x0 ,t0 ) < ∞
Mt
implies that Mt has locally finite H n -measure, which is all we need for
the integral formulation of the mean curvature flow (6.3). A solution
(Mt )t∈I of the mean curvature flow (6.1) satisfies this assumption for
any t ∈ I if we require that the flow has polynomial area-growth when
you intersect the flow with sufficiently large balls, that is to say
Area(Mt ∩ B(x0 , R)) ≤ ARp ,
for all t ∈ I and R >> 0,
(see [Eck04, Lemma C.3].)
This inequality can be obtained due to an appropriate condition on
initial hypersurface. Indeed, if for instance I = (0, T ) and M0 satisfies
Area(M0 ∩ B(x0 , R)) ≤ ARp
for all R ≥ R0 , for certain R0 big enough, then the local estimation of
Proposition 6.9 implies
Area(Mt ∩ B(x0 , R)) ≤ 2p+3 ARp
√
for each t ∈ (0, T ) as long as R ≥ max{ 2nT , R0 /2}.
The proof given in [Eck04] of the monotonicity formula also produces
the following extension involving weight functions that (jointly with
MEAN CURVATURE FLOW
63
their derivatives) are integrable on Mt with respect to Φ(x0 ,t0 ) . In particular, these functions with compact support defined on a properly
embedded solution (6.1) form an important family. This extension is
the base of a local version of the formula of monotonicity ( Proposition
6.23), which is a fundamental tool in regularity theory in the first singular moment. Below we state the specific result.
Theorem 6.17 (Weighted Monotonicity Formula). Let (Mt )t∈I
be a family of surfaces moving by the mean curvature and fix (x0 , t0 ) ∈
Rn+1 ×R. Assume f is a smooth function (probably depending on time)
define on (Mt )t∈I satisfying:
Z ∂f
2
|f | + | | + |Df | + |D f | Φ(x0 ,t0 ) < ∞
∂t
Mt
for all t ∈ I con t < t0 . Then, for t < t0 , we have
2 Z Z
⊥
∇
Φ
d
d
(x
,t
)
0
0
~ −
f Φ(x0 ,t0 ) .
f Φ(x0 ,t0 ) =
− ∆Mt f − H
dt Mt
dt
Φ
(x
,t
)
Mt
0 0
Moreover, if f is non-negative and verifies:
d
− ∆Mt f ≤ 0,
dt
then
d
dt
Z
f Φ(x0 ,t0 ) ≤ 0.
Mt
Proof. The demonstration of this theorem follows the same steps as in
Theorem 6.15. The only difference is that we use the function f · φ
instead of φ as a test function in formula (6.20).
Remark 6.18. Representation formula for the standard heat equation.
Let Mt = Rn for any t < t0 and consider f = f (x, t) a solution of the
standard heat equation in Rn . Theorem 6.17 yields
Z
d
f (x, t)Φ(x0 ,t0 ) (x, t)dx = 0
dt Rn
for all t < t0 . Indeed,
Z
Z ∇⊥ Φ(x0 ,t0 ) 2
d
d
~
f Φ(x0 ,t0 ) =
− ∆Mt f − H −
f Φ(x0 ,t0 ) =
dt Rn
dt
Φ(x0 ,t0 ) Rn
Z ∇⊥ Φ(x0 ,t0 ) 2
=
0 − 0 −
f Φ(x0 ,t0 ) = 0
Φ(x0 ,t0 ) Rn
64
FRANCISCO MARTIN AND JESUS PEREZ
where we are using that
d
dt
~ = 0 and that
− ∆Mt f = 0, that H
∇⊥ Φ(x0 ,t0 ) R= 0.
Therefore Rn f (x, t)Φ(x0 ,t0 ) (x, t)dx does not depend on t for t < t0 . On
the other hand, as f is continuous, we have that
Z
lim
f (x, t)Φ(x0 ,t0 ) (x, t)dx = f (x0 , t0 ),
t%t0
Rn
Now we will discuss some consequences of the monotonicity formula.
Theorem 6.19 (Upper Bound for the Area Ratio). Consider a smooth,
properly embedded MCF, Mt , in B(x0 , r0 ) × (t0 − r02 , t0 ). Then
Area(Mt ∩ B(x0 , r))
≤
rn
t∈(t0 −r02 ,t0 )
sup
κ(n) ·
!
for any r ∈
Area(Mt0 −r02 /(2(2n+1)) ∩ B(x0 , r0 ))
r0n
r0
0, p
.
2(2n + 1)
Proof. Take a =
p
1/(2(2n + 1)) and consider
f = ϕ(x0 ,t0 −a2 r02 ),r0 .
Fix r ∈ (0, a r0 ). Since we have that
d
− ∆Mt f ≤ 0,
dt
then we can apply the Weighted Monotonicity Formula to the
function f and the point (x0 , t0 + r2 ) ∈ Rn+1 × R.
So, for t ∈ (t0 − r02 , t0 ), we have:
Z
Φ(x0 ,t0 +r2 ) · ϕ(x0 ,t0 −a2 r02 ),r0 dµt ≤
Mt
Z
Mt
Φ(x0 ,t0 +r2 ) · ϕ(x0 ,t0 −a2 r02 ),r0 dµt
2 2
0 −a r0
For the right-hand side we use that:
• spt(ϕ(x0 ,t0 −a2 r02 ),r0 ) ⊆ B(x0 , r0 ).
• |ϕ(x0 ,t0 −a2 r02 ),r0 | ≤ 1 and |Φ(x0 ,t0 +r2 ) | ≤
1
an r0n
MEAN CURVATURE FLOW
65
For the left-hand side we use that, for r ∈ (0, a r0 ), one has:
ϕ(x0 ,t0 −a2 r02 ),r0 ≥
and
Φ(x0 ,t0 +r2 ) ≥
1
8
1
e1/4 (8π)n/2 rn
on B(x0 , r) × (t0 − r2 , t0 ). This completes the proof.
In order to refer more easily to the solutions of the mean curvature
flow we will often denote M = (Mt )t∈I .
An immediate consequence of the non-positivity of the right-hand side
of the Monotonicity Formula (Theorem 6.15) is the following proposition (see [Eck04]):
Proposition 6.20 (Gaussian Density). Let M = (Mt )t∈(t1 ,t0 ) be a
solution of the MCF satisfying the assumptions of Theorem 6.15. Then
for any x0 ∈ Rn+1 the Gaussian density
Z
(6.21)
Θ(M , x0 , t0 ) = lim
Φ(x0 ,t0 )
t%t0
Mt
exists.
The solutions of the mean curvature flow which are homothetically contracting with respect to (x0 , t0 ), i.e., satisfying (see [Eck04, p. 10-13])
(x − x0 )⊥
~
H(x)
=
2(t − t0 )
for all x ∈ Mt y t < t0 , can be also characterized by the property:
Z
(6.22)
Θ(M , x0 , t0 ) =
Φ(x0 ,t0 )
Mt
for all t < t0 .
Remark 6.21. It is clear from its definition that, for any t ∈ (t1 , t0 ),
Z
Θ(M , x0 , t0 ) ≤
Φ(x0 ,t0 ) .
Mt
Example 6.22. (A) Our first example is the case of hyperplane P ⊂
Rn+1 . Taking into account that:
• the Hausdorff measure is invariant under isometries of Euclidean space,
66
FRANCISCO MARTIN AND JESUS PEREZ
• the measure H m in Rm coincides with the standard Lebesgue
measure m-dimensional,
• the Riemann integral of the heat kernel on Rn is 1,
then
Z
(6.23)
Z
n
Φ(x0 ,t0 ) (x, t)dH (x) =
Φ(x)dx = 1
Rn
P
for all t < t0 . Therefore,
Z
(6.24)
Θ(M , x0 , t0 ) = lim
t%t0
Φ(x0 ,t0 ) = lim 1 = 1
Mt
t%t0
where M = P because a hyperplane is minimal and therefore it is
stationary.
(B) A. Stone computed in [Sto94] the density for other flows like shrinkm
ing spheres and cylinders. We consider Mt := Sn−m
r(t) × R , r(t) =
p
n−m
2(m − n)t, t < 0. It is clear that M−1/2 = S√
× Rm , so by
n−m
(6.22) in the above proposition we have
n−k
Z
2
e−|x| /2
n−k 2
n−m
√
Θ(M , 0, 0) =
>1
dµ
=
·
Area
S
n−m
n/2
2π e
M−1/2 (2π)
Thus, for the surfaces in Euclidean 3-space we have
p that Θ(M , 0, 0) =
4/ e in the case of cylinders and Θ(M , 0, 0) = (2π)/ e then M
consists of concentric spheres.
6.1. Parabolic rescaling. In the general case, the density value can
be calculated using parabolic scale changes of the solution M = (Mt )t∈I
in the way we are going to describe in this subsection. We consider
x ∈ Mt and make the following change of variable:
x = λy + x0
t = λ2 s + t0
where λ > 0 and s < 0. Then
1
y ∈ (Mλ2 s+t0 − x0 ) ≡ Ms(x0 ,t0 ),λ
λ
and for a fix λ > 0
(Ms(x0 ,t0 ),λ )s<0
is solution of the MCF equation.This will show at the end of this section
for not diverting our attention from the main objective of studying the
Gaussian density following a parabolic scaling.
MEAN CURVATURE FLOW
67
Figure 5. Shrinking cylinders.
The integral of the Gaussian density can be calculated by applying a
change of variable:
Z
Z
|x−x0 |2
1
− 4(t −t)
0
e
dH n (x) =
Φ(x0 ,t0 ) =
n/2
[4π(t0 − t)]
Mt
ZMt
Z
|y|2
1
n
4s
=
e dH (y) =
Φ.
n/2
(x ,t ),λ
1
(Mλ2 s+t −x0 ) (−4πs)
Ms 0 0
λ
0
Therefore for any s < 0,
Z
(6.25)
Θ(M , x0 , t0 ) = lim
t%t0
Z
Φ(x0 ,t0 ) = lim
Mt
λ&0
(x0 ,t0 ),λ
Ms
Φ.
68
FRANCISCO MARTIN AND JESUS PEREZ
In case the point (x0 , t0 ) ∈ Rn+1 × R verifies x0 ∈ Mt0 , then
lim Ms(x0 ,t0 ),λ = Tx0 Mt0
λ&0
for all s < 0 (smoothly on compact sets). Taking this into account and
using (6.25) and (6.24), we get
Z
(6.26)
Θ(M , x0 , t0 ) =
Φ(y, s)dH n (y) = 1.
Tx0 Mt0
(x ,t ),λ
Finally, as we promised, we want to see that (Ms 0 0 )s<0 is a solution of the MCF equation. Indeed, we know that Mt moves by the mean
curvature, that is, the re exits a smooth embedding Ft = F (·, t) : M →
~
Rn+1 with Mt = Ft (M ) and such that ∂F
(x, t) = H(x,
t) ∀t ∈ I.
∂t
(x0 ,t0 ),λ
To check that (Ms
)s∈J is also solution it suffices to prove that
n+1
Gs = G(·, s) : M → R
given by
G(p, s) =
1
F (p, λ2 s + t0 )
λ
verifies:
∂
G(p, s) = H~G (p, s),
∂s
~ G is the mean curvature of G, which is H
~ G = λ · H.
~ Hence,
where H
the equality we are looking for is just an easy consequence of the chain
rule.
6.2. Local monotonicity formula. In this subsection we are going
to use again the test functions:
3
|x − x0 |2 + 2n(t − t0 )
ϕ(x0 ,t0 ),ρ (x, t) = 1 −
ρ2
+
that were already introduced in Remark 6.7. We used them in the
proof of Proposition 6.9 (local area bound.)
Take (Mt )t∈(t1 ,t0 ) a family of hypersurfaces moving by the mean curvature inside a domain U ⊂ Rn+1 . Fix x0 ∈ U , there exists ρ0 > 0 so
that
√
B(x0 , 1 + 2n ρ0 ) × (t0 − ρ20 , t0 ) ⊂ U × (t1 , t0 ).
For each ρ ∈ (0, ρ0 ) and t ∈ (t0 − ρ20 , t0 ) we have that that the support
of our test function satisfies:
√
(6.27)
spt ϕ(x0 ,t0 ),ρ (·, t) ⊂ B(x0 , 1 + 2n ρ0 ) ⊂ U.
MEAN CURVATURE FLOW
69
Proposition 6.23 (Local Monotonicity Formula). Let (Mt )t∈(t1 ,t0 )
n+1
a smooth mean curvature
√ flow in U ⊂ R . Then for each x0 ∈ U
there exists a ρ0 ∈ (0, t0 − t1 ) such that for all ρ ∈ (0, ρ0 ] and t ∈
(t0 − ρ2 , t0 )
spt ϕ(x0 ,t0 ),ρ (·, t) ⊂ U
and
(6.28)
2
Z
Z d
(x − x0 )⊥ ~
Φ(x0 ,t0 ) ϕ(x0 ,t0 ),ρ ≤ −
H(x) − 2(t − t0 ) Φ(x0 ,t0 ) ϕ(x0 ,t0 ),ρ .
dt Mt
Mt
Since the right-hand side of (6.28) is negative and ϕ(x0 ,t0 ),ρ (x0 , t0 ) = 1
for each ρ ∈ (0, ρ0 ], then the following local Gaussian density
Z
(6.29)
Θ(M , x0 , t0 ) = lim
Φ(x0 ,t0 ) ϕ(x0 ,t0 ),ρ
t%t0
Mt
is well defined, is independent of ρ and for global solutions of MCF
equation it coincides with the density defined in Proposition 6.20.
Furthermore, for all t ∈ (t0 − ρ2 , t0 )
Z
Φ(x0 ,t0 ) ϕ(x0 ,t0 ),ρ .
(6.30)
Θ(M , x0 , t0 ) ≤
Mt
Proof. In order to get 6.28 we only have to apply (6.20) for φ = ϕ(x0 ,t0 ),ρ ,
taking into account that ϕ(x0 ,t0 ),ρ satisfies (6.11).
For the second part of the proof we are going to use some ideas due
to B. White [Whi97]. For x0 ∈ U and ρ > 0 such that
B(x0 , 2ρ) × (t0 −
ρ2
, t0 ) ⊂ U × (t1 , t0 ).
2n
By Proposition 6.9 we deduce
sup
2
ρ
(t0 − 2n
,t0 )
Area (Mt ∩ B(x0 , ρ)) ≤ 8 Area Mt
ρ2
0 − 2n
∩ B(x0 , 2 ρ) ≡ k0 .
Consider ψρ ∈ C02 (B(x0 , ρ)) satisfying: χB(x0 ,ρ/2) ≤ ψρ ≤ χB(x0 ,ρ) and
|D2 ψρ | ≤ k1 , where c1 depends on ρ. Then
d
dt − ∆Mt ψρ ≤ k2 · χB(x0 ,ρ)−B(x0 ,ρ/2) ,
for a suitable constant k2 depending on n and k1 . On B(x0 , ρ) −
B(x0 , ρ/2) we have Φ(x0 ,t0 ) ≤ k3 , where k3 depends on n and ρ. Then
70
FRANCISCO MARTIN AND JESUS PEREZ
the weighted monotonicity formula gives that
Z
d
ψρ Φ(x0 ,t0 ) ≤ k4 ,
dt Mt
where
k4 depends on the previous constants. So, the function t 7→
R
ψ Φ
− c4 t is non-increasing and therefore the following limit
Mt ρ (x0 ,t0 )
exists
Z
ψρ Φ(x0 ,t0 ) .
lim
t%t0
Mt
It is possible to check that this limit is independent of ρ [Whi97]. and
hence it does not depend on the particular function ψρ which is constantly 1 in an neighborhood of x0 . So, (6.29) makes sense. Inequality
(6.30) is consequence of the monotonic behavior of the integral.
Remark 6.24 (Local monotonicity under re-scaling). Changing
the scale in U × (t1 , t0 ) as we made in Remark 6.21, then a smooth,
properly embedded, mean curvature flow (Mt )t∈(t1 ,t0 ) becomes
in λ−1 (U − x0 ) × (λ−2 (t1 − t0 ), 0).
(M (x0 ,t0 ),λ )
s
s∈ λ−2 (t1 −t0 ),0
√
If ρ0 is such that B(x0 , 1 + 2nρ0 ) ⊂ U , then for any ρ ∈ (0, ρ0 ] we
have:
Z
(6.31)
Θ(M , x0 , t0 ) = lim
Φ(y, s)ϕλ−1 ρ (y, s)dH n (y)
λ&0
(x0 ,t0 ),λ
Ms
for all s < 0, where
3
2
2 |y| + 2ns
ϕλ−1 (y, s) = 1 − λ
ρ2
+
is the re-scaled test function.
In particular, as we made in Remark 6.21, we can deduce
(6.32)
Θ(M , x0 , t0 ) = 1
for all x0 ∈ Mt0 .
One of the main applications of the local monotonicity is the study
of the limits of mean curvature flow after changes in scale. The next
result, which concludes this section, goes in this direction. It is an immediate consequence of Propositions 6.20 and 6.23 and Remark 6.24.
Proposition 6.25 (Tangent flows/ Parabolic Blow-ups). Let M =
(Mt )t∈(t1 ,t0 ) be a smooth, properly embedded, mean curvature flow in a
domain U ⊂ Rn+1 . Let x0 ∈ U and assume that for a given sequence
MEAN CURVATURE FLOW
71
(x ,t ),λ
−2
{λj }j∈N & 0 the hypersurfaces (Ms 0 0 j ) in λ−1
j (U −x0 )×(λj (t1 −
t0 ), 0) obtained as in Remark 6.21 converge16, smoothly on compact
sets of Rn+1 , to a properly embedded solution of the MCF equation
M 0 = (Ms0 )s<0 . Then M 0 satisfies
Z
0
(6.33)
Θ(M , 0, 0) =
Φ = Θ(M , x0 , t0 )
Ms0
for all s < 0. Therefore, from Theorem 6.15, we deduce
(6.34)
y⊥
~
H(y)
=
2s
for all y ∈ Ms0 and s < 0. Moreover, this means that
√
0
(6.35)
Ms0 = −s · M−1
for each s < 0.
Definition 6.26. The limit M 0 = (Ms0 )s<0 is called tangent flow or
parabolic “blow-up” of M in (x0 , t0 ).
It is posible to show (6.35) from (6.34), for instance by proving directly from the definition of MCF that the hypersurfaces (Ms0 ) verify
∂ F (φ(p, s), s)
√
=0
∂s
−s
where φ(·, s) : M → M is a family of diffeomorphisms satisfying
>
∂φ
∂F
dF
=−
.
∂s
∂s
16Given
(Mj )j∈N ⊂ U and M ⊂ U properly embedded hypersurfaces in
U ⊂ Rn+1 , we say that (Mj , gj )j∈N smoothly converge to (M, g) in U if M is
the pointwise limit of (Mj )j∈N and for all p ∈ M there exist r, > 0 such that
(1) M ∩ W (p, r, ) can be writen as the graph of a function u : D(p, r) → R.
(2) For each j (large enough) the hypersurface Mj ∩ W (p, r, ) can be also
written as the graph of a function uj : D(p, r) → R and {uj } converges to
u in the topology of C k convergence on compact subsets of D(p, r), for any
k ∈ N,
Here we have used the following notation: D(p, r) = {p + v : v ∈ Tp M, |v| < r}
means the tangent disk of radius r > 0 and W (p, r) = {q+tν(q) : q ∈ D(p, r), t ∈ R}
is the solid cylinder of radius r around the normal line determined by the Gauß
map of M at p; ν(p). Inside W (p, r), given > 0, we take the compact cylinder
W (p, r, ) = {q + tN (q) : q ∈ D(p, r), |t| < }.
72
FRANCISCO MARTIN AND JESUS PEREZ
√
This means that F (φ(p,s),s)
is a family of embeddings that does not
−s
depend on s. As at the instant s = −1 we know that
F (φ(·, −1), −1)
0
p
,
= F (φ(·, −1), −1) = M−1
−(−1)
√
0
0
0
√
then F (φ(·,s),s)
=
M
for
all
s
<
0,
that
is,
M
=
−sM−1
, as we
−1
s
−s
wanted to prove.
7. Some Remarks About Singularities
Throughout this section, we consider a compact initial hypersurface
M . Consider T maximal such that a smooth solution of the MCF
F : M × [0, T ) → Rn+1 as in Theorem 2.6 exists. Then the embedding
vector F is uniformly bounded according to Corollary 5.10. Then some
spatial derivatives of the embedding Ft have to become unbounded as
t % T . Otherwise, we could apply Arzelà-Ascoli Theorem and obtain
a smooth limit hypersurface, MT , such that Mt converges smoothly to
MT as t % T . This is impossible because, in such a case, we could
apply Theorem 2.6 to re-start the flow. In this way, we could extend
the flow smoothly all the way up to T + ε, for some ε > 0 small enough,
contradicting the maximality of T . In particular, we have that |h|2 is
not bounded, when we approach the maximal time T .
We would like to say more about the “blowing-up” of the norm of h,
as t % T. Recall that , according to (3.9), the evolution equation for
|h|2 is
2
∂ 2
|h| = ∆|h|2 − 2∇h + 2|h|4 .
∂t
Label
|h|2max := max |h|2 (·, t).
Mt
Using Hamilton’s trick (Lemma 5.6) we deduce that |h|2max is locally
Lipschitz and that
∂
d
|h|2max (t0 ) =
|h|2 (p0 , t0 ),
dt
∂t
where p0 is any point where |h|2 (·, t0 ) reaches its maximum. Thus,
using the above expression, we have
d
∂
|h|2max (t0 ) =
|h|2 (p0 , t0 ) =
dt
∂t
2
∆|h|2 (p0 , t0 ) − 2∇h(p0 , t0 ) + 2|h|4 (p0 , t0 )
MEAN CURVATURE FLOW
73
It is well known that the Hessian of |h| is negative semi-definite at
any maximum. In particular the Laplacian of |h| at these points is
non-positive. Hence,
d
|h|2max (t0 ) ≤ 2|h|4 (p0 , t0 ) ≤ 2|h|4max (t0 ).
dt
Notice that |h|2max is always positive, otherwise at some instant t we
would have that h ≡ 0, along Mt , which would imply that Mt is a
hyperplane Rn+1 , which is contrary to the fact that the initial data is
a compact hypersurface.
So, one can prove that 1/|h|2max is locally Lipschitz. Then the previous
inequality allows us to deduce that:
d
1
−
≤ 2, a.e. in t ∈ [0, T ).
dt |h|2max
Integrating (respect to time) in any sub-interval [t, s] ⊂ [0, T ) we get
1
1
−
≤ 2(s − t).
2
|h(·, t)|max |h(·, s)|2max
As h is not bounded as to tends to T , then there exists a time sequence
si % T such that
|h(·, si )|2max → +∞.
Substituting s = si in the above inequality and taking limit, as i → ∞,
we get
1
≤ 2(T − t).
|h(·, t)|2max
We collect all this information in the next proposition.
Proposition 7.1. Consider the mean curvature flow for compact initial hypersurface M . If T is the maximal time of existence, then the
following lower bound holds
1
max |h(p, t)| ≥ p
p∈M
2(T − t)
for all t ∈ [0, T ).
In particular,
lim max |h(p, t)| = +∞.
t→T p∈M
Definition 7.2. When this happens we say that T is singular time for
the mean curvature flow.
74
FRANCISCO MARTIN AND JESUS PEREZ
So we have the following improved version of Theorem 2.6:
Theorem 7.3. Given a compact, immersed hypersurface M in Rn+1
then there exists a unique mean curvature flow defined on a maximal
interval [0, Tmax ).
Moreover, Tmax is finite and
1
max |h(p, t)| ≥ p
p∈M
2(Tmax − t)
for each t ∈ [0, Tmax ).
Remark 7.4. From the above proposition, we deduce the following
estimate for the maximal time of existence of flow:
1
.
Tmax ≥
2|h(·, 0)|2max
Definition 7.5. Let T be the maximal time of existence of the mean
curvature flow. If there is a constant C > 1 such that
C
max |h(p, t)| ≤ p
,
p∈M
2(T − t)
then we say that the flow develops a Type I singularity at instant T .
Otherwise, that is, if
p
lim sup max |h(p, t)| (T − t) = +∞,
t→T
p∈M
we say that is a Type II singularity.
We conclude this brief section by pointing out that there have been
substantial breakthroughs in the study and understanding of the singularities of type I, whereas type II singularities have been much more
difficult to study. This seems reasonable since, according to the above
definition and the results we have seen, the singularities of type I are
those for which has the best possible control of “blow-up” of the second
fundamental form.
References
[And94]
[Ber60]
B. Andrews, Contraction of convex hypersurfaces in Riemannian
spaces, Journal of Differential Geometry 39 (1994), 407–431.
M. Berger, Les variétés Riemanniennes 1/4-pincées, Ann. Scuola
Norm. Sup. Pisa 14 (1960), 161–170.
MEAN CURVATURE FLOW
[Bra78]
75
K.A. Brakke, The motion of a surface by its mean curvature, Princeton
University Press, 1978.
[Bra01]
H. Bray, Proof of the Riemannian Penrose inequality using the positive
mass theorem, Journal of Differential Geometry 59 (2001), 177–267.
[BS09]
S. Brendle and R. Schoen, Manifolds with 1/4-pinched curvature are
space forms, J. Amer. Math. Soc. 22 (2009), no. 1, 287–307.
[DC92]
M.P. Do Carmo, Riemannian geometry, Springer, 1992.
[Cha93]
I. Chavel, Riemannian geometry - A modern introduction, Cambridge
University Press, 1993.
[CDR03]
U. Clarenz, G. Dziuk, and M. Rumpf, On generalized mean curvature
flow in surface processing, S. Hildebrandt, H. Karcher (Eds.): Geometric Analysis and Nonlinear Partial Differential Equations, 217-248,
Springer Verlag, 2003.
[CM11]
T.H. Colding and W. Minicozzi, A course on minimal surfaces, Graduate Studies in Mathematics, vol. 121, AMS, 2011.
[DHKW92] U. Dierkes, S. Hildebrandt, A. Küster, and O. Wohlrab, Minimal surfaces I, Boundary Value Problems, Grundlehren der mathematischen
Wissenschaften, vol. 295, Springer-Verlag, 1992.
[Eck95]
K. Ecker, On regularity for mean curvature flow of hypersurfaces, Calculus of Variations and PDE’s 3 (1995), 107–126.
, A local monotonicity formula for mean curvature flow, Annals
[Eck01]
of Mathematics 154 (2001), 503–525.
[Eck04]
, Regularity Theory for Mean Curvature Flow, Birkhauser, 2004.
[ES91]
L.C. Evans and J. Spruck, Motion of level sets by mean curvature. I,
Journal of Differential Geometry 33 (1991), no. 3, 635–681.
[GH86]
M.E. Gage and R. S. Hamilton, The heat equation shrinking convex
plane curves, Journal of Differential Geometry 23 (1986), no. 1.
[Ger73]
R. Geroch, Energy Extraction, Ann. New York Acad. Sci. 224 (1973),
108–117.
[Gra87]
M. A. Grayson, The heat equation shrinks embedded plane curves to
round points, Journal of Differential Geometry 26 (1987), no. 2.
[GP74]
V. Guillemin and A. Pollack, Differential Topology, Prentice Hall, 1974.
[HE73]
S. Hawking and G. Ellis, The Large Structure of Space-Time, Cambridge Monographs on Mathematical Physics, Cambridge University
Press, 1973.
[Hui90]
G. Huisken, Asymptotic behaviour for singularities of the mean curvature flow, Journal of Differential Geometry 31 (1990), 285–299.
[HI01]
G. Huisken and T. Ilmanen, The Inverse Mean Curvature Flow and the
Riemannian Penrose Inequality, Journal of Differential Geometry 59
(2001), no. 3, 353–437.
[Ilm95]
T. Ilmanen, Lectures on Mean Curvature Flow and Related Equations,
Preprint, 1995.
[JW77]
P.-S. Jang and R. Wald, The positive energy conjecture and the cosmic
censor hypothesis, J. Math. Phys. 18 (1977), no. 1, 41–44.
[Kli61]
W. Klingenberg, Über Riemannsche Mannigfaltigkeiten mit positiver
Krümmung, Comment. Math. Helv. 35 (1961), 47–54.
[Lie96]
G.M. Lieberman, Second Order Parabolic Differential Equations, World
Scientific Publishing Company, 1996.
76
[Man11]
[MSHS14]
[MIP12]
[Mor00]
[MS13]
[Rau51]
[Nad96]
[Pen73]
[Pen82]
[RS10]
[SS14]
[SY79]
[Smo11]
[Sto94]
[Tay96]
[Whi97]
[Whi13]
FRANCISCO MARTIN AND JESUS PEREZ
C. Mantegazza, Lectures Notes on Mean Curvature Flow, Birkhäuser,
2011.
F. Martı́n, A. Savas-Halilaj, and K. Smoczyk, On the topology of translating solitons of the mean curvature flow, preprint (2014).
W.H. Meeks III and J. Pérez, A survey on classical minimal surface
theory, University Lecture Series (AMS) vol. 60, 2012.
F. Morgan, Geometric Measure Theory. A beginner’s guide (Third Edition), Academic Press, 2000.
O. Müller and M. Sánchez, An Invitation to Lorentzian Geometry,
Jahresbericht der Deutschen Mathematiker-Vereinigung (2013), 1–31.
H.E. Rauch, A contribution to differential geometry in the large, Ann.
of Math. 54 (1951), 38–55.
N. Nadirashvili, Hadamard’s and Calabi-Yau’s conjectures on negatively curved and minimal surfaces, Invent. Math. 126 (1996), no. 3,
457–465.
R. Penrose, Naked singularities, Ann. New York Acad. Sci. 224 (1973),
125–134.
, Some unsolved problems in classical general relativity, Ann.
Math. Study 102 (1982), 631–668.
M. Ritoré and C. Sinestrari, Mean Curvature Flow and Isoperimetric Inequalities, Advanced Courses in Mathematics. CRM Barcelona,
Birkhäuser Verlag, 2010.
M. Sáez and O.C. Schnürer, Mean curvature flow without singularities,
Journal of Differential Geometry 97 (2014), no. 3, 545–570.
R. Schoen and S.-T. Yau, On the proof of the positive mass conjecture
in General Relativity, Comm. Math. Phys. 65 (1979), no. 1, 45–76.
K. Smoczyk, Mean curvature flow in higher codimension - Introduction and survey, Global Differential Geometry, Springer Proceedings in
Mathematics, 2012, Volume 17, Part 2, 231-274, 2011.
A. Stone, A density function and the structure of singularities of the
mean curvature flow, Calculus of Variations and PDE’s 2 (1994), 443–
480.
M.E. Taylor, Partial Differential Equations III: Nonlinear Equations,
volume 117 of Applied Mathematical Sciences, Springer, 1996.
B. White, Stratification of minimal surfaces, mean curvature flows
and harmonic maps, Journal reine und angewandte Math. 488 (1997),
1–35.
, Topological Change in Mean Convex Mean Curvature Flow,
Invent. math. 191 (2013), 501–525.
MEAN CURVATURE FLOW
Francisco Martı́n
Departmento de Geometrı́a y Topologı́a
Universidad de Granada
18071 Granada, Spain
E-mail address: fmartin@ugr.es
Jesús Pérez
Departmento de Geometrı́a y Topologı́a
Universidad de Granada
18071 Granada, Spain
E-mail address: perez.garcia.jesus@gmail.com
77