Research Statement
Xin Zhou
My main research interests focus on Differential Geometry and General Relativity, and their relation to
the Calculus of Variations, Topology, and Probability Theory. I am particularly interested in minimal surface theory and geometric inequality/rigidity questions arising from General Relativity. One main subject
of my recent work focuses on the min-max theory of minimal surfaces and its applications to Topology and
Gaussian probability space; my research on General Relativity focuses on the mass angular momentum inequality for rotating black holes. My research mainly uses geometric measure theory and partial differential
equations.
I will give a brief introduction to my recent results in Part I, elaborate on these results in Part II, and
propose some future research in Part III.
Part I:
Introduction to results
(I) Min-max theory, its application and Gaussian probability space
One very common phenomenon in minimal surface theory is that it is by no means easier to get the desired geometric properties of the solution, although the existence and regularity theory is already extremely
deep and hard. My research in this part solves several significant problems pertaining to this phenomenon.
(1) Geometric characterization of Almgren-Pitts min-max hypersurface, [74, 76].
Around 1970s, F. Almgren and J. Pitts developed a Morse theoretical method for constructing minimal
hypersurfaces, namely the min-max theory. This deep and powerful theory had been almost silent until its
recent celebrated application by F. Marques and A. Neves to solve the Willmore conjecture [35]. There are
many questions left open in this field. One important conjecture, raised by Almgren [3], is to understand the
geometric properties of the constructed min-max hypersurface, such as the Morse index, multiplicity and
the size of area. In [74, 76], I answer this question for manifolds with positive Ricci curvature. In particular,
I characterize the Morse index, area and multiplicity of this min-max hypersurface:
The Almgren-Pitts min-max hypersurface is either orientable and of index one, or is a double cover of
a non-orientable stable minimal hypersurface. The min-max hypersurface has least area among all closed
embedded minimal hypersurfaces (counting the non-orientable case with multiplicity two).
My result is the first instance that addresses the geometry of the Almgren-Pitts min-max hypersurface in
general dimensions, and has many influences on other areas. Recently L. Mazet and H. Rosenberg used my
idea to give an application to minimal surface theory in hyperbolic 3-manifolds [37]. With Y. Liokumovich
[44], we can use my result to prove a generalization of Gromov’s systolic inequality; see part (3). Besides
these applications, the ideas and tools developed here have fundamental significance for the further study of
min-max theory in high dimensions.
(2) Entropy and min-max theory in Gaussian probability space, [33].
In [33] with D. Ketover, we establish a min-max theory for minimal surfaces in the Gaussian probability
|x|2
1 − 4
space (R3 , 4π
e
dx2 ); as an application, we solve a conjecture in Mean Curvature Flow (MCF).
Entropy is a natural geometric quantity measuring the complexity of a surface Σ embedded in R3 as
introduced by T. Colding and W. Minicozzi [18]. Specifically, the entropy λ(Σ) is the supremum of the area
1
under the Gaussian metric over all scalings λΣ (λ > 0) and translations Σ + x (x ∈ R3 ) of this surface. For
dynamical reasons relating to MCF, Colding-Ilmanen-Minicozzi-White [12] conjectured that the entropy of
any closed surface is at least that of the self-shrinking two-sphere (a special solution of MCF in R3 ).
In [33], we prove this conjecture for all closed embedded 2-spheres using our min-max theory in the
Gaussian space. Assuming a conjectural Morse index bound (announced recently by Marques-Neves), we
can improve the result to apply to all closed embedded surfaces that are not tori.
Our results can be thought of as the parabolic analog to the Willmore conjecture, and our argument is
analogous in many ways to that of Marques-Neves on the Willmore problem [35]. This is also the first case
beyond the Willmore Conjecture where the min-max method and index/genus classification theorems can
give lower bounds for area of minimal surfaces. Our min-max theory applied to the Gaussian area in R3 is
the first instance of a global variational theory in a non-compact incomplete manifold. We can also use the
idea and method to establish min-max theory on other non-compact spaces. The min-max theory itself will
have many applications in the Gaussian probability space.
(3) Application to a generalization of Gromov’s systolic inequality, [44].
The systole of a closed Riemannian manifold (M n , g) is the length of the shortest non-contractible
geodesic loop. Gromov’s systolic inequality [28] says that the systole is bounded by Cvol(M )1/n for a
large class of manifolds with a universal constant C. In order to generalize the systolic inequality to more
general settings, L. Guth [31] asked whether for every Riemannian metric g on a 3-manifold M it is possible
1
to construct a map f : (M, g) → R2 with fibers of length at most Cvol(M, g) 3 .
In [44], with Y. Liokumovich, we affirmatively answer this question for 3-manifolds M under an assumption of positive Ricci curvature. In particular, we can construct a sweepout of M by 1-cycles of length
at most Cvol(M 3 , g)1/3 . These 1-cycles are the fibers of a smooth function f : M → R2 .
Our result is a strong generalization of the systolic inequality. It is to our knowledge the first occasion
where such a generalization is proved. Moreover, our result can recover the classical and several other
generalized versions of the systolic inequality.
(4) Existence of minimal surfaces of arbitrarily large Morse index, [42].
In [42] with H. Li, we prove that in a closed 3-manifold with a generic metric of positive Ricci curvature,
there are minimal surfaces of arbitrarily large Morse index, and this confirms a conjecture by Marques and
Neves [36].
Aiming at answering a conjecture by S. T. Yau [68, Problem 88], Marques-Neves applied the AlmgrenPitts min-max theory to canonical k-parameter families of hypersurfaces introduced by Gromov [29] and
Guth [30], and proved the existence of infinitely many closed, embedded, minimal hypersurfaces in manifold
of positive Ricci curvature [36]. In [36], they further conjectured that in a generic metric the Morse indices
of these minimal hypersurfaces should grow linearly. However, since the argument of Marques-Neves is
by contradiction and hence implicit, it is really hard to get any information of these minimal hypersurfaces directly. In fact, it is totally open before our work that whether the Morse indices of these minimal
hypersurfaces will grow up to infinity; our result is the first affirmative answer toward this conjecture.
(II) General Relativity
(5) Mass angular momentum inequality, [62, 73].
2
In this project, we study the relation between mass and angular momentum in General Relativity. In
particular, we prove that the angular momentum is bounded from above by the total mass in an appropriate
way for a large class of rotating, vacuum black hole, and Einstein-Maxwell black hole initial data sets. This
inequality is automatically true for the Kerr and Kerr-Newman black hole solutions which are stationary.
For dynamical, axisymmetric solutions some general results have been obtained, first by S. Dain [20] and
later by other authors [6, 8, 7, 19], by assuming the solution is maximal.
With R. Schoen [62], we introduce a new method for obtaining such inequalities which is technically
simpler and provides sharper results in many cases. In particular, we weaken the asymptotic conditions to
the optimal ones, and we also give a quantitative version of such inequalities.
In a sequel paper [73], I obtain such inequalities for non-maximal, rotating, vacuum black holes, by
assuming the trace is small.
The maximal condition is conjectured by physicists to be unnecessary. Our result is the only affirmative
answer in the non-maximal case up to now. A very new perspective is introduced where both elliptic and
hyperbolic PDE methods are used together to acquire the result.
(III) Min-max theory via mapping method
(6) On the existence of min-max minimal surfaces of genus ≥ 1, [71, 72].
I establish a min-max theory for minimal surfaces using sweepouts of surfaces of genus g ≥ 1 via the
method of harmonic maps. As a result, I prove that the min-max value for the area functional is achieved by
a bubble tree limit consisting of a branched, noded, genus-g minimal surface, and finitely many branched
minimal spheres. I also obtain a Colding-Minicozzi type strong convergence theorem.
The existence theory for unstable genus-g minimal surfaces was initiated by Sacks-Uhlenbeck using
perturbation method [53]. The flow method was studied by Ding-Li-Liu (g = 1) [21] and Topping, Rupflin,
et al. (g ≥ 2) [49, 50, 51]. Recently a viscosity method was introduced by T. Rivière [48]. Compared
to these works, my result has several significant advantages. The perturbation method by Sacks-Uhlenbeck
may have possible energy loss, and hence can not in general achieve a bubble tree with area equal to the minmax value. Although the flow method has no energy loss, the area of the limit can not be a priori prescribed
to be the min-max value. The viscosity method of Rivière does not provide as good an approximating
sequence as mine, which is very important for geometric applications.
(7) On free boundary min-max geodesics, [75].
I establish a general min-max theory for constructing geodesics with free boundary. Given a Riemannian
manifold M with a constraint submanifold N , we construct geodesics with free boundary lying on N via
min-max method.
When the submanifold N is empty our result reduces to the famous Birkhoff curve shortening process
[15, Chap 5, §2]. Our existence result can be viewed as a full generalization of the previous results by A.
Weinstein [65] (in a standard ball with a Finsler metric) and A. Nabutovsky-R. Rotman [41] (for geodesic
loops where N is a point), in the sense that the total space M and the constraint N can be any manifold and
submanifold. The ideas behind this can be used to establish the min-max theory for minimal surfaces with
free boundary.
3
Part II:
Description of main results
Research on the Almgren-Pitts min-max theory
1
Geometric characterization of Almgren-Pitts min-max hypersurface
Minimal surfaces are critical points of the area functional in a given Riemannian manifold, and they
play an important role in the study of the geometry and topology of the ambient manifolds. A natural way
to produce minimal surfaces is to minimize area within a homology class. This idea leads to the famous
existence and regularity theory for area minimizing hypersurfaces by De Giorgi, Federer, Fleming, Almgren,
Simons, etc. (cf. [22, 63]). In general, when every hypersurface is homologically trivial, the minimization
method fails. This motivated Almgren [2, 3] and Pitts [45], to develop a Morse theoretical method for the
area functional on the space of hypersurfaces, namely the min-max theory. In particular, given a closed
Riemannian manifold (M n+1 , g), consider a continuous one-parameter family Φ : [0, 1] → Zn (M ), where
Zn (M ) denotes the space of all closed hypersurfaces in M . Let
[Φ] be the set of all Ψ : [0, 1] → Zn (M )
which are homotopic to Φ in Zn (M ). If [Φ] 6= 0 in π1 Zn (M ) , then Φ is called a sweepout. The min-max
value can be associated with [Φ] as
L([Φ]) = inf max Area(Ψ(x)) : Ψ ∈ [Φ] .
(1.1)
x∈[0,1]
Pl
In [3, 45], it is proven that when 2 ≤ n ≤ 5, L([Φ]) =
i=1 ki Area(Σi ) for a disjoint collection
(nonempty) of closed, embedded minimal hypersurfaces {Σi }li=1 with multiplicity ki ∈ N. The case when
n ≥ 6 is due to Schoen and Simon [56]; Σi has a singular set of codimension 7 when n ≥ 7.
However, this deep and powerful theory had been almost silent until recently due to two issues. One
issue is the lack of geometric information, e.g. Morse index and multiplicity, known about these minmax hypersurfaces. The other issue is the super-complicated discrete machinery used by Almgren-Pitts,
which makes it hard to apply to geometric objects. This situation has changed recently. Marques and
Neves introduced a new point of view to study the Morse index of the min-max surface in three dimension
[34], and successfully proved the famous Willmore conjecture [35] by combining the index estimate with
a discretization procedure, which works as a bridge to connect continuous geometric objects to the discrete
machinery used by Almgren-Pitts. While the argument for the Morse index bound of Marques-Neves is
restricted to dimension three, I [74, 76] develop several new ideas and succeed in characterizing significant
geometric properties, including the Morse index, multiplicity, orientation, and area, of the Almgren-Pitts
min-max hypersurface in all dimensions.
To be precise, given a minimal hypersurface Σ in M , the Morse index of Σ is the number of negative
eigenvalues of the Jacobi operator associated with the second variation of area. Denote S to be the class
of all closed, embedded, minimal hypersurfaces (possibly with a singular set of codimension 7) in M . By
[45, 56], S =
6 ∅. Let
n Area(Σ), if Σ is orientable
WM = inf
.
2Area(Σ), if Σ is non-orientable
Σ∈S
In [74] (2 ≤ n ≤ 6) and [76] (n ≥ 7), we prove that:
Theorem 1.1. If the Ricci curvature of M is positive, then the min-max minimal hypersurface Σ is:
4
(i) either orientable of multiplicity one, has Morse index Ind(Σ) = 1, and Area(Σ) = WM ;
(ii) or non-orientable with multiplicity two, and stable, i.e. Ind(Σ) = 0, and 2Area(Σ) = WM .
One interesting aspect of our result says that the min-max hypersurface has least area among all minimal hypersurfaces (if one counts non-orientable ones with multiplicity two). The existence of a least area
minimal surface in a 3-manifold is a starting point for Marques-Neves in [34, 35]. However, in general
dimensions, nothing about the existence of a least area minimal hypersurface is known before our result. To
tackle this problem, several new ideas are introduced which are described in the following.
The main idea in low dimensions 2 ≤ n ≤ 6 [74] is as follows. Given Σ ∈ S, Σ can be embedded
into a canonical “smooth” family (level sets of a Morse function) {Σt }t∈[0,1] , where a unique maximum of
{Area(Σt )}t∈[0,1] is achieved by either Area(Σ) when Σ is orientable, or by 2Area(Σ) when Σ is nonorientable. Then we show
that for any Σ ∈ S the canonical family {Σt }t∈[0,1] lies in the same homotopy
class in π1 Zn (M ), {0} in the sense of Almgren-Pitts. Therefore if Σ0 is the min-max hypersurface, the
canonical family associated with Σ0 is of a “mountain pass” type, or equivalently, there is no other ways
to deform the area of Σ0 down besides the direction along the canonical family. This obviously implies
that Σ0 has Morse index one. A major difficulty comes from two different topologies used in the space of
hypersurfaces Zn (M ); the canonical family is continuous under a very weak topology (the flat topology),
while the Almgren-Pitts theory uses a much stronger topology (the mass norm topology). In order to convert
the geometric family to that used by Almgren-Pitts, a discretization argument is used essentially as in the
proof of the Willmore conjecture by Marques-Neves.
In higher dimensions n ≥ 7 [76], several new key ingredients need to be developed to deal with the
presence of singularities. First given a singular minimal hypersurface Σ, there is no way to embed Σ into a
smooth family as above due to singularities. Instead, we use the level sets of the distance function from Σ
as the canonical family. Using an idea by Gromov [27] and volume comparison results, we can show that
this distance family is of the “mountain pass” type. However, the price we pay for using this non-smooth
family is revealed in the second step, i.e. to show that all the distance families associated with singular
hypersurfaces lie in the same homotopy class. In fact, our distance family does not a priori satisfy a key
assumption of Marques-Neves’ discretization theorem, i.e. the no mass concentration condition. Actually
this condition is conjectured by Marques-Neves to be unnecessary in [35]. In [76], as an essential technical
tool, we prove a stronger discretization theorem. Our result removes the no mass concentration condition,
and hence confirms the conjecture of Marques-Neves. In particular, we prove that:
Theorem 1.2. Given a continuous (under the flat topology) 1-parameter family of hypersurfaces Φ :
[0, 1] → Zn (M ), such that for each x ∈ [0, 1], Φ(x) is represented by the boundary of some set Ωx ⊂ M
of finite perimeter, and such that maxx∈[0,1] Area Φ(x) < ∞, then there exists a discrete sequence {φi }
which is continuous (under the mass norm topology) in the sense of Almgren-Pitts, satisfying
max Area Φ(x) = lim sup max Area(φi (x)).
x∈[0,1]
i→∞
x∈[0,1]
The key step is to develop a new discretization procedure to connect the given family to a new family
which satisfies the no mass concentration condition, and this turns out to be a highly non-trivial process.
2
Entropy and min-max theory in Gaussian probability space
2.1 Entropy conjecture: The resolution of Willmore Conjecture by Marques-Neves [35] hinges on proving that in the round S3 , the Clifford torus is the non-equatorial embedded minimal surface with smallest
5
area. In [33] with D. Ketover, we address the analogous question for singularity models for the mean curvature flow (MCF) and cast the question in terms of min-max theory.
R −|x|2 /4
1
For a smooth surface Σ ⊂ R3 , its Gaussian area is F (Σ) = 4π
dx. Critical points for
Σe
Gaussian area are called self-shrinkers and arise as blowup limits at singularities of MCF [14]. Following
Colding-Minicozzi [14], the entropy of Σ is the supremum of Gaussian areas over all translations (t ∈ R3 )
and rescalings (s ∈ [0, ∞)) of Σ:
λ(Σ) := sup F (s(Σ − t)).
(2.1)
t,s
The entropy of a self-shrinker is equal to its Gaussian area, analogous to the fact that the Willmore energy
is equal to area for a minimal surface in S3 . The self-shrinker of smallest area is the flat plane, and one
can ask (as Marques-Neves [35] in S3 ) which non-flat self-shrinker has smallest area above the flat plane.
Using dynamical methods, Colding-Ilmanen-Minicozzi-White [12] proved that the compact self-shrinker
with smallest area above the plane is the self-shrinking 2-sphere S2 . They further conjectured:
Conjecture 2.1. [12] For any smooth closed embedded surface Σ in R3 ,
λ(Σ) ≥ λ(S2 ).
(2.2)
Our main result is a min-max proof of this conjecture for closed 2-spheres:
Theorem 2.2. Let Σ ⊂ R3 be a smooth closed embedded 2-sphere. Then
λ(Σ) ≥ λ(S2 ).
Assuming a conjectural index bound for min-max limits that has recently been announced by MarquesNeves, Theorem 2.2 can be extended to all closed surfaces with genus not equal to 1.
We now briefly sketch the argument for Theorem 2.2. Starting with a smooth two-sphere Σ, we consider
the canonical 4-parameter family of genus 0 surfaces of translates and dilates of Σ given by Σs,t = s(Σ − t).
This is a natural sweepout since translates and dilates are the 4 unstable directions that are always present
for any self-shrinker. By definition, the entropy λ(Σ) is greater than or equal to the Gaussian area of any
surface in this family. The entropy thus gives an upper bound for the width of this canonical homotopy class
of sweepouts (see (2.3) for definition). The min-max theory that we develop for Gaussian area (§2.2) then
gives a self-shrinker Γ of area at most λ(Σ) and genus 0. By the classification of Brendle [4] this shrinker
must be the plane, sphere or cylinder (the entropy of a cylinder is larger than that of a sphere). The crucial
fact is that just as in the proof of the Willmore Conjecture, the boundary of our sweepout records the genus
of Σ, and we can then use a topological degree argument to rule out getting the plane. Thus we can show
Γ = S2 and hence obtain F (S2 ) = λ(S2 ) ≤ λ(Σ).
For a general surface (not necessary genus 0), the Morse index of the resulting self-shrinker Γ should be
at most 4 (by Marques-Neves’ announcement), and must be the plane, sphere or cylinder by the classification
of Colding-Minicozzi [14], so that all other arguments follow the same way.
2.2
Min-max theory in Gaussian space: Since self-shrinkers are unstable minimal surfaces of the Gaus|x|2
1 − 4
sian probability space (R3 , 4π
e
dx2 ) [14, §0.2], any variational construction of self-shrinkers must be
of the min-max type. In [33], we establish a min-max theory in the Gaussian space using the framework of
the Simon-Smith min-max method [64, 9].
Let us first briefly introduce the setup. Given n ∈ N, denote I n = [0, 1]n and ˚
I n = (0, 1)n . We consider
n-parameter family of surfaces in R3 which are continuous in the following sense.
6
Definition 2.3. A family {Σν }ν∈I n of smooth surfaces in R3 is said to be a continuous (genus g) family, if
• {Σν } is a smooth family under the locally smooth topology;
• For t ∈ (0, 1)n , Σν is a genus g surface, and Σν is an affine plane or empty set when ν ∈ ∂I n ;
• F (Σν ) is a continuous function of ν, and supν∈I n F (Σν ) < ∞.
Given a continuous family {Σν }ν∈I n , we can generate new continuous families by deforming the Σν by
isotopies. In particular, take a map ψ ∈ C ∞ (I n × R3 , R3 ) such that ψ(ν, ·) is an isotopy for each ν ∈ I n ,
and define {Σ0ν } by Σ0ν = ψ(ν, Σν ). Such a family {Σ0ν } is said to be homotopic to {Σν }.
Given a homotopy class Λ of such n-parameter families, the min-max value of the Gaussian area F is
W (Λ) = inf
maxn F (Σν ) .
(2.3)
{Σν }∈Λ
ν∈I
The main min-max theorem we prove in [33] is the following:
Theorem 2.4. For any homotopy class Λ, if W (Λ) > maxν∈∂I n F (Σν ), then there is a connected, smooth,
embedded, Gaussian minimal surface with Gaussian area W (Λ) (counted with multiplicity) and with genus
at most g.
Now we describe the key idea of our strategy. The only key ingredient in the Simon-Smith theory [64]
(see also [9]) which can not be adapted easily to the Gaussian space is the so-called “tightening” process,
which is a pseudo-gradient flow of the area functional on the space of varifolds (c.f. [63, §38]). The subtlety
stems from that the space of varifolds in R3 with bounded Gaussian area is no longer compact. In particular,
a sequence of surfaces may weakly converge to a limit surface together with a point mass at infinity. To
overcome this difficulty, we compactify R3 by adding a point at infinity to get the three sphere S3 , and view
all the varifolds in R3 with bounded Gaussian area as varifolds in S3 . Then we design a specific pseudogradient flow of the Gaussian area F on the space of varifolds in S3 . Our flow will either push a varifold
to be F -stationary in R3 , or decrease the mass near infinity. After applying this flow, all the sequences of
surfaces of our interests will converge to a stationary varifold (with respect to the Gaussian metric) with no
point mass at infinity, and hence fulfill our requirement for constructing the “tightening” process.
3
Application to a generalization of Gromov’s systolic inequality
Let M be a 3-manifold with positive Ricci curvature. In [44] with Y. Liokumovich, we obtain quantitative results about sweepouts of M by 1-cycles and surfaces.
Theorem 3.1. Every closed 3-manifold M of positive Ricci curvature admits a sweepout by 1-cycles of
1
length bounded above by Cvol(M ) 3 for a universal constant C > 0.
This result can be used to prove the following systolic inequality.
Theorem 3.2. Let M be a closed 3-manifold of positive Ricci curvature not homeomorphic to a 3-sphere.
1
Then M contains a non-contractible closed geodesic of length at most Cvol(M ) 3 .
Theorem 3.2 extends Gromov’s systolic inequality [28] to the case of 3-mainfolds with non-trivial fundamental group and positive Ricci curvature. If M is topologically a sphere, then a min-max argument
yields an upper bound for the length of a stationary 1-cycle, giving an alternative proof of a special case of
the result of Nabutovsky and Rotman [40].
We also obtain sweepouts of M by surfaces with controlled area and genus.
7
Theorem 3.3. Given a three manifold M of positive Ricci curvature, there exists a minimal surface Σ20 with
Area(Σ0 ) ≤ Cvol(M )2/3 for a universal constant C > 0. Also we have
• If Σ0 is orientable, then the genus g0 of Σ satisfies g0 ≤ 3, and there exists a smooth sweepout of M
by 2-surfaces {Σt }t∈[−1,1] with g(Σt ) = g0 and Area(Σt ) ≤ Area(Σ0 ).
• If Σ0 is non-orientable, then the genus g̃0 of its double cover Σ̃0 satisfies g˜0 ≤ 3, and there exists
a smooth sweepout of M̃ = M \ Σ0 by 2-surfaces {Σt }t∈[0,1] with g(Σt ) = g0 and Area(Σt ) ≤
2Area(Σ0 ).
In [34], assuming Ricg > 0 and scalar curvature Scalg ≥ 6, Marques-Neves produce a smooth sweepout
{Σt }t∈[0,1] of M , where the genus of Σt is the Heegaard genus, and Area(Σt ) ≤ 4π. Although they have
better genus estimates, our area estimates can be much better while we still have a good genus estimate.
The main idea of proving Theorem 3.1 is a dimension reduction argument. Particularly, we continuously
sweep out these 2-surfaces Σt in Theorem 3.3 by 1-cycles with length bounded by CArea(Σt )1/2 . Theorem
3.3 is proved by combining several ingredients. We apply the Almgren-Pitts min-max theory to the sweepout
constructed in [26] and get a min-max minimal surface of controlled area. By using my Morse index bound
[74] (see also §1), we can get the desired genus bound via Schoen-Yau genus estimates [69]. The existence
of good sweepouts by surfaces follows from my work in [74] and Meeks-Simon-Yau [38].
4
Existence of minimal surfaces of arbitrarily large Morse index
In this project, we confirm a conjecture by Marques-Neves [36] in dimension three. Let M 3 be a 3manifold. A metric g on M is called bumpy, if any closed embedded minimal surface has no nontrivial
Jacobi field. Bumpy metrics are generic by White [66]. In [42] with H. Li, we prove the following:
Theorem 4.1. Let M 3 be a closed 3-manifold with a bumpy metric g of positive Ricci curvature. Then there
exist embedded minimal surfaces of arbitrarily large Morse index in (M 3 , g).
The problem of finding minimal surfaces with arbitrarily large Morse index in 3-manifolds has attracted
a lot of interest since a well-known question of Pitts and Rubinstein (c.f. [46, 13]). Motivated by this
question, a smooth metric was constructed on any 3-manifold which admits embedded minimal surfaces of
arbitrarily large Morse index by Hass-Norbury-Rubinstein [32] (genus zero), Colding-Hingston [11] (genus
1) and Colding-De Lellis [10] (genus g ≥ 2). Compared to these results, we produce a large family of
metrics on a given 3-manifold, which admit minimal surfaces of arbitrarily large Morse index.
In fact, Theorem 4.1 is a direct corollary of the following finiteness theorem for minimal surfaces of
bounded Morse index in a bumpy metric and Marques-Neves’ result in [36]. Let M 3 be a closed 3-manifold
with a bumpy metric g of positive Ricci curvature, and M the space of closed, embedded, minimal surfaces
in (M, g). By [36], M contains infinitely many elements. Given N 1, we denote MN by
MN = {Σ ∈ M :
Ind(Σ) ≤ N }.
In [42], we prove
Theorem 4.2. There are only finitely many elements in MN .
8
(4.1)
We briefly sketch the proof of Theorem 4.2 as follows. The idea is to study the lamination convergence of
a sequence of minimal surfaces of bounded Morse index. If the theorem were not true, then take a sequence
{Σi } in MN . The Morse index bound implies that the Σi are locally stable away from finitely many points,
and hence converge locally smoothly to a minimal lamination L outside these points by Schoen’s curvature
estimates [54]. Assume that L is orientable and consider the leaves of L. When Λ is an accumulating leaf
or an isolated leaf for which the multiplicity of convergence of Σi to Λ is greater than one, we construct a
positive Jacobi field along Λ, and hence show that Λ has only removable singularities; so its closure Λ is a
smooth, embedded, stable minimal surface, which contradicts the positive Ricci curvature condition [23]. In
the case of an isolated leaf with multiplicity one convergence, we construct a nontrivial Jacobi vector field
which extends across the singularities, hence contradicting the bumpy condition. For the case that L is nonorientable, we can proceed similarly by combining the above arguments with the structure of non-orientable
surfaces (c.f. [74, §3]).
Research on mathematical General Relativity
5
Mass angular momentum inequality
In General Relativity, asymptotically flat manifolds are used to model isolated galaxy and black hole
systems. Physical quantities of such systems, such as the ADM mass and linear momentum, have been
defined and studied in a purely geometric way. Motivated by the gravitational collapse picture, geometric
inequalities involving these quantities, such as the positive mass theorem by Schoen-Yau [60, 61] and Witten
[67], have attracted a lot of interest. Recently, a new type of such inequalities, called mass angular momentum inequalities, which bound the angular momentum by the ADM mass, were introduced by Dain [20],
and later by other authors [6, 8, 7, 19] for a class of simply connected, asymptotically flat, axisymmetric,
maximal, vacuum and electro-vacuum black hole data of the Einstein and Einstein/Maxwell equations.
5.1 Quantitative results: In [62] with R. Schoen, we introduce a new method to obtain such inequalities
with sharper results. Therefore we can generalize the asymptotic conditions in [20, 6, 8, 7, 19] to the optimal
ones, and we also acquire a quantitative inequality (5.2) measuring the L6 -distance between general data
and the extreme Kerr and Kerr-Newman black hole solutions, which characterize the rigidity cases.
To be precise, let (M, g, k) be an asymptotically flat manifold with two ends, where g is a Riemannian
metric and k is a symmetric 2-tensor, with the optimal asymptotic decay conditions:
1
3
gij = δij + os (r− 2 ), kij = Os−1 (r−λ ) : s ≥ 5, λ > .
2
Such a triple (M, g, k) models the initial data set for black hole solutions of the Einstein equations.
Assume that (g, k) satisfies the vacuum constraint equations, and is maximal, i.e. trg k = 0. Suppose
that (g, k) is axisymmetric under an axial Killing vector field. By [6] when π1 (M ) = 0, such (M, g, k)
determines a singular map to the hyperbolic plane (U, w) : R3 \ {0} → (H2 , ds2−1 ), and the ADM mass m
of (M, g) is bounded from below by the reduced energy I(U, w), where
Z
I(U, w) =
|DU |2 + ρ−4 e4U |Dw|2 dx.
(5.1)
R3
In fact, the map (U0 , w0 ) corresponding
to the extreme Kerr data with angular momentum J is a local critical
p
point of I, with I(U0 , w0 ) = |J|. Consider the class of maps (U, w) determined by (M, g, k) as above
with fixed angular momentum J. In [62], we prove:
9
Theorem 5.1. I(U,
pw) is bounded from below by I(U0 , w0 ) among all data satisfying the above requirement. Hence m ≥ |J|. Moreover,
Z
√
d6−1 (U, w), (U0 , w0 ) dx}1/3 ,
(5.2)
m − J ≥ I(U, w) − I(U0 , w0 ) ≥ C{
R3
where d−1
(U, w), (U0 , w0 ) is the hyperbolic distance on (H2 , ds2−1 ).
The main idea is to show that the reduced energy I is convex among singular maps (U, w) with the same
asymptotic boundary behaviors, motivated by a classical convexity estimate of the harmonic energy by R.
Schoen [55]. In [62], we also obtain mass angular momentum/charge inequality for axisymmetric maximal
Einstein/Maxwell black hole data with an L6 -norm bound between the given data and the extreme KerrNewman data. As a result, we give the first proof of the strict mass angular momentum/charge inequality
when the given data is not identical with the extreme Kerr-Newman solution.
5.2 Non-maximal case: All the previous results [20, 6, 8, 62] require the data to be maximal, i.e. trg k =
0, which is unnecessary by the gravitational collapse pictures. In [73], I prove the mass angular momentum
inequality for non-maximal, axisymmetric, vacuum black hole data by assuming the trace trg k is small.
In particular, motivated by Schoen-Yau’s proof of the spacetime positive mass theorem [61], we construct
an axisymmetric, maximal data with the same mass and angular momentum as the given one when trg k is
small, then the desired inequality follows from Theorem 5.1.
To be precise, let (Σ, e) be a 3-dimensional manifold which is Euclidean at infinity. Denote Hs,δ (Σ) to
δ+|α| α
be the weighted Sobolev space containing functions with norm kuk2Hs,δ (Σ) = Σ|α|≤s k|x|e
D uk2L2 (Σ,e) <
∞, where α are multi-indices. Denote VC s+1,δ+ 1 (Σ) and MVC s+1,δ+ 1 (Σ) by the set of vacuum data and
2
2
maximal vacuum data (g, k) respectively, such that (g − e, k) ∈ Hs+1,δ+ 1 (Σ) × Hs,δ+ 3 (Σ).
2
2
When (Σ0 , e) is simply connected and axisymmetric with two ends, we can assume that Σ0 ' R3 \ {0},
and the axisymmetric Killing field η is the rotation vector field around z-axis. Denote MVC as+1,δ (Σ0 ) to be
the subset of MVC s+1,δ (Σ0 ) of data which are symmetric under η. In [73], we prove:
Theorem 5.2. Given an axisymmetric data (g, k) ∈ VC as+2,δ+ 1 (Σ0 ) with s ≥ 7, − 32 < δ < −1, there
2
is an > 0, depending only on the norm of (g, k), such that if ktrg kkH
MVC as+1,δ (Σ0 ),
3
s−2,δ+ 2
(Σ)
≤ , then we can
with the same ADM mass m and
construct an axisymmetric maximal data (gu , ku ) ∈
angular momentum J as that of (g, k). As a corollary of Theorem 5.1 (applied to (Σ0 , gu , ku )), we have
p
m ≥ |J|.
We briefly sketch the proof as follows. First we construct a boosted spacetime satisfying the vacuum
Einstein equations evolving from the given black hole data set (g, k). A key step is to adapt ChristodoulouO’Murchadha’s boost evolution theory [5] from R3 to an asymptotically flat end. When the trace is small,
a perturbation method is used to solve the maximal surface equation in the boosted spacetime with certain
growth condition at infinity. The setup of our perturbation method is very novel, as we take the whole
initial data set (g, k) and a graphical function u as input and the mean curvature of the graph H(g, k, u)
inside the boosted spacetime as output. The linearization of this map with respect to the graphical function
Du H(g, k, u) is surjective thanks to the vacuum Einstein equations. When the initial data set (g, k) is
axisymmetric, we can make the maximal graph (gu , ku ) axisymmetric with the same angular momentum as
that of (g, k), and hence prove Theorem 5.2.
10
Our method has general interest in the study of deformation theorems in General Relativity. For example,
it can be used to prove the mass angular momentum/charge inequality for non-maximal, rotating, EinsteinMaxwell black hole initial data sets with small trace.
Research on the min-max theory via mapping method
6
On the existence of min-max minimal surfaces of genus ≥ 1
Conformal and harmonic maps from two-dimensional domains to a Riemannian manifold parametrize
two-dimensional minimal surfaces. The study of minimal surfaces from this perspective originated from the
celebrated proof of classical Plateau problem by Douglas and Rado in 1930s. Much remarkable research
has been done afterward, e.g. Schoen-Yau’s study of three manifolds with nonnegative scalar curvature
[60], and Micallef-Moore’s proof of the topological sphere theorem [39]. Among them, the min-max theory
was initiated by combining a perturbation argument with classical Morse theory by Sacks and Uhlenbeck
[52, 53], and was then explored extensively by many others. Recently, another remarkable work concerning
min-max theory was done by Colding and Minicozzi [14, 16], where they constructed min-max minimal
spheres via harmonic map theory, and used them to prove the finite time extinction of 3-dimensional Ricci
flow, and hence gave a proof of the last step towards the famous Poincaré conjecture. The idea of ColdingMinicozzi is to generalize Birkhoff’s curve shortening process (cf. [17, Chap 5]) to two dimensions, and a
significant feature is a strong convergence result (similar to Theorem 6.1) compared to the method of SacksUhlenbeck, which plays an essential role in estimating the decay rate of the width under the Ricci flow.
In [71, 72], I establish the min-max theory for higher genus minimal surfaces, and also achieve a strong
convergence result. The increased genus makes the theory even more sophisticated than [16], and several
novel ideas pertaining to the Teichmüller theory are introduced.
Let us describe the results in more precise terms. Given a Riemannian manifold (M, h) and a closed
surface Σ0 of genus g ≥ 1. Like Morse theory, we are interested in the saddle points of the area
func0
1,2
tional in C ∩ W (Σ0 , M ). In particular, consider a nontrivial path β(t) in the path space Ω = γ(t) ∈
C 0 [0, 1], C 0 ∩ W 1,2 (Σ0 , M ) . The min-max critical value associated with the homotopy class [β] in Ω is
defined in [71, 72] by,
W = inf max Area ρ(t) .
ρ∈[β] t∈[0,1]
In order to use the variational method in C 0 ∩ W 1,2 to get the critical points associated with W, we need
to work with the harmonic energy functional E instead of the area functional. Since the energy functional
depends notonly on the map, but also on the conformal structure of Σ0 , we introduce thecoupled path
space Ω̃ = (γ(t), τ (t)) : γ(t) ∈ C 0 [0, 1], C 0 ∩ W 1,2 (Στ (t) , M ) , τ (t) ∈ C 0 ([0, 1], Tg ) , where Tg is
the Teichmüler space on a genus g surface (Ω̃ reduces to Ω when Σ0 is a sphere since T0 is a point). The
min-max critical value of the energy functional is defined in [71, 72] by,
WE =
inf
max E ρ(t), τ (t) .
(ρ,τ )∈[(β,τ0 )] t∈[0,1]
In fact we prove that W = WE . The existence and strong convergence results I prove in [71, 72] can be
summarized together as follows.
Theorem 6.1. Given β ∈ Ω, if W > 0, there exists a minimizing sequence (ρn , τn ) ∈ [β], with
11
maxt∈[0,1] E ρn (t), τn (t) → W, and for any > 0, there exists a large integer N > 0 and a small δ > 0
such that if n > N , and for any t ∈ (0, 1) satisfying:
E ρn (t), τn (t) > W − δ,
(6.1)
there are a conformal harmonic map u0 : Σg → M defined on the body Σg of a genus g surface with nodes
Σ∗g and finitely many harmonic spheres {ui : S 2 → M }li=1 , such that:
dV ρn (t), ∪ui ≤ ,
(6.2)
i
where dV is the varifold distance as in [16, Appendix A].
Heuristically the theorem says that “almost maximal (6.1) implies almost critical (6.2)”.
The proof of Theorem 6.1 contains several steps analogous to the proof of the classical Plateau problem.
The first step is to show that W = WE . It is a consequence of the following strong uniformization result,
which says that a continuous family of metrics on a given surface uniquely determines a continuous family
of elements in the Teichmüller space and a continuously family of conformal maps. It is the first instance
where a uniformization result works continuously for a family of metrics. In particular, we prove:
Theorem 6.2. Given a C 1 metric g on Σ0 , there is a unique τ ∈ Tg with a representative Στ , and a unique
orientation preserving C 1,1/2 conformal isotopy h : Στ → (Σ0 , g). Moreover, if {g(t) : t ∈ [0, 1]} is a
continuous family of C 1 metrics, then the corresponding pairs (τ (t), h(t)) are continuous with respect to
the parameter t in Tg and C 0 ∩ W 1,2 (Στ (t) , Σ0 ) respectively.
This is proved by studying a PDE governing the quasi-conformal maps [71, Appendix] motivated by the
method of Ahlfors-Bers [1]. Various representation of the Teichmüller space are entangled together. We
apply this result to make our minimizing sequences of paths in Ω̃ almost conformal, such that the area and
harmonic energy of each map among these sequences are uniformly close.
Afterward we adapt Colding-Minicozzi’s local harmonic replacement process (a two-dimensional analog of Birkhoff’s curve shortening process) to the higher genus case in a highly non-trivial way, where we
need a delicate treatment of the change of domain surfaces. This process leads to compactness of every
min-max sequence in the sense of Sacks-Uhlenbeck’s bubbling convergence process. The convergence has
no energy loss due to the fact that the limits are conformal (i.e. W = WE ).
7
On the free boundary min-max geodesics
In [75], I establish a min-max theory for geodesics with free boundary. Let (M, g) be a closed Riemannian manifold, and N a closed submanifold. A geodesic segment of M which meets N orthogonally
is called a free boundary geodesic. We work in the Sobolev space
W 1,2 (I, M ), whereI = [0, 1] and the
R
W 1,2 -norm of a map f : I → M ⊂ RN is given by kf k2W 1,2 = [0,1] |f (x)|2 + |f 0 (x)|2 dx. The energy of
R
f is defined by E(f ) = [0,1] |f 0 (x)|2 dx.
The total variational space is Ω = {σ(t) ∈ C 0 [0, 1], W 1,2 (I, M ) : σ(t, 0), σ(t, 1) ∈ N }. Given
σ0 ∈ Ω, denote [σ0 ] by the homotopy class of σ0 in Ω. The width associated to [σ0 ] is defined by:
W0 = W [σ0 ] = inf max E σ(t) .
(7.1)
σ∈[σ0 ] t∈[0,1]
Our main result says that the width is always achieved by a free boundary geodesic.
12
Theorem 7.1. There exists a free boundary geodesic γ with E(γ) = W ([σ0 ]).
When π1 (M, N ) is non-trivial, there always exists σ0 with W ([σ0 ]) > 0, and hence a nontrivial free
boundary geodesic. Theorem 7.1 is a direct corollary of the following Theorem 7.2. It is a ColdingMinicozzi type strong min-max result (see [15]), which says that “almost maximal implies almost critical”.
Theorem 7.2. There exists a sequence {γj }j∈N ⊂ [σ0 ], with limj→∞ maxt∈[0,1] E γj (t) = W0 , such that,
for any > 0, there exists a δ > 0, and if j > 1δ , and if for some t0 ∈ [0, 1]
E γj (t0 ) > W0 − δ,
then dist(γj (t0 ), G) < , where G is the space of free boundary geodesics.
The main idea is to adapt the Birkhoff’s curve shortening process (BCSP) to manifold with a constraint
submanifold. Briefly, given a closed curve γ with 2L evenly spaced break points {x0 , x1 , · · · , x2L = x0 } ⊂
S 1 , the BCSP first replaces each piece γ|[x2i ,x2i+2 ] on even intervals by a geodesic segment connecting
γ(x2i ) with γ(x2i+2 ), and then repeats the geodesic replacement process on odd intervals [x2i−1 , x2i+1 ]. In
our case, given a curve γ : [0, 1] → M , γ(0), γ(1) ∈ N , with 2L + 1 evenly spaced break points {x0 =
0, x1 , · · · , x2L = 1} ⊂ [0, 1], we first replace the boundary piece γ|[0,x2 ] (and γ|[x2L−2 ,1] ) by a geodesic
segment γ̃ connecting γ(x2 ) (and γ(x2L−2 )) to the constraint N , then do replacements on inner pieces
γ|[x2i ,x2i+2 ] , i 6= 0, L − 1 and γ|[x2i−1 ,x2i+1 ] as BCSP. Our modified BCSP satisfies properties analogous
to BSCP. Among them, one key ingredient is to show that the W 1,2 -norm difference kγ|[0,x2 ] − γ̃kW 1,2 is
controlled by the length difference Length(γ|[0,x2 ] ) − Length(γ̃). This is achieved by a convexity estimate,
which will also be useful in other free boundary variational problems.
Part III:
Future research plan
(1) Almgren-Pitts min-max theory and applications.
I would like to continue investigating the Almgren-Pitts theory based on [74, 76, 44, 42]. One aim is
to understand the geometry of the Almgren-Pitts min-max hypersurface in general manifolds. So far in
[74, 76], we focused on manifolds with positive Ricci curvature, where stable minimal hypersurfaces—
an important obstruction toward understanding the min-max theory, do not exist by [24, 76]. In general
manifolds, we propose to cut the ambient manifold along stable minimal hypersurfaces, and then discuss
the min-max theory for manifolds with boundary.
In [44], we mainly obtained the 2-width estimate for 1-cycles in 3-manifold. An analogous problem
as Guth did in [30] is to bound the k-width for 1-cycles by the volume when k ≥ 3. Based on [44], Y.
Liokumovich and I will approach this problem by cutting the 3-manifold into small balls and again using
min-max theory for manifolds with boundary.
Concerning the min-max hypersurfaces originating from multi-parameter families by Marques-Neves
[36], so far my work with H. Li [42] is restricted to 3 dimensions. In the future, I plan to pursue the study
of the Morse index and area growth of these min-max hypersurfaces in general dimensions.
The current research in this field only focuses on the codimention one case. I would like to explore the
min-max theory in high codimension cases. In particular, I am interested in establishing a min-max theory
for Lagrangian minimal surfaces as in Schoen-Wolfson [58]. My other expertise on the mapping theory in
[71, 72] can also help on this problem.
(2) Min-max theory in manifolds with boundary.
13
In an ongoing project with Martin Li, we are developing an Almgren-Pitts min-max theory in a manifold
M with non-empty boundary ∂M , aiming at finding minimal surfaces which meet ∂M orthogonally. These
are called free boundary minimal surfaces, which are critical points of the area functional among properly
immersed surfaces Σ with ∂Σ ⊂ ∂M . In fact, the general setup of Almgren [3] covers the free boundary
case, but the regularity issue at the free boundary was untouched. With Martin Li, we can prove a full
boundary regularity of the solution without any restriction on the boundary ∂M . We have proved the free
boundary version of the maximum principle and curvature estimates of Schoen-Simon-Yau [57]. After that,
we will formulate an “outer almost minimizing property” as [43] in order to obtain the desired regularity.
In the future, we will study the geometric properties of these free boundary min-max surfaces and their
application to detect the geometry and topology of the ambient manifold.
(3) Min-max theory in Gaussian space and other smooth metric measure spaces.
First I would like to explore the application of the min-max theory in Gaussian space established in
[33], for instance its relation to probability theory and problems in mean curvature flow beyond the entropy
conjecture. Besides the Gaussian space, I will study the min-max theory in general non-compact smooth
metric measure spaces (e.g. gradient shrinking Ricci solitons). The goal is to construct a complete and
non-compact weighted minimal surface. The key ingredient is to construct a suitable vector field which can
deform the area of a surface down if the area accumulates near infinity.
(4) Min-max theory via harmonic maps.
The Morse index bound for the min-max minimal surfaces constructed by Colding-Minicozzi and myself
[16, 71, 72] is an important open problem that I plan to address in the future. In fact, the Morse index bound
for min-max minimal spheres by the perturbation method in [52] was known, and played an essential role in
the topological sphere theorem by Micallef and Moore [39]. I will approach this problem by combining our
strong convergence result with the idea of Micallef-Moore. Moreover, I wish to answer a question of Yau
[70, Problem 30] on the branch points by relating them to the Morse index bound.
It is interesting to establish the min-max theory via harmonic maps for free boundary minimal surfaces.
Fraser [25] adapted the Sacks-Uhlenbeck’s perturbation method to find free boundary min-max minimal
disks. I plan to build up the corresponding theory using Colding-Minicozzi’s harmonic replacement process.
Furthermore, it is also interesting to study the corresponding theory in the Gaussian probability space.
(5) Mathematical General Relativity.
I would like to pursue the study of the mass angular momentum inequalities based on our works [62, 73],
aiming at removing the small trace condition in [73]. Until now we mainly focused on the single black hole
case. Our method in [62] is very likely applicable to the multi-black hole case which is still open. Another
interesting problem is whether the inequality holds for black holes with physically meaningful matter fields.
This is not consistent with the Newtonian physics, and I plan to construct a counterexample.
Regarding the geometric inequality/rigidity problems, I plan to study the positive mass theorem in high
dimensions which has been used quite often in conformal geometry as a hypothesis. R. Schoen has proposed
a proof using singular minimal hypersurfaces. My expertise on singular minimal hypersurfaces hopefully
can help to finish this program.
I am also interested in the theory of maximal hypersurfaces in a Lorentzian spacetime—a spacetime
analog of minimal hypersurfaces. The existence theory of such hypersurfaces has mainly been approached
by the PDE method in the past. I would like to pursue a variational theory toward constructing such hypersurfaces. Also, I wish to study the relation between the formulation of singularities in spacetime and the
14
foliation of spacelike constant mean curvature hypersurfaces.
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