S URFACES OF C ONSTANT M EAN
C URVATURE
C ARL J OHAN L EJDFORS
Master’s thesis
2003:E11
CENTRUM SCIENTIARUM MATHEMATICARUM
Centre for Mathematical Sciences
Mathematics
i
Abstract
The aim of this Master’s dissertation is to give a survey of some basic results
regarding surfaces of constant mean curvature (CMC) in R 3 . Such surfaces
are often called soap bubbles since a soap film in equilibrium between two
regions is characterized by having constant mean curvature. The surface area
of these surfaces is critical under volume-preserving deformations.
CMC surfaces may also be characterized by the fact that their Gauss map
N : ! S 2 is harmonic i.e. it satisfies
S
S
t
t(N ) = 0;
where (N ) is the tension field of N , generalizing the classical Laplacian. This
is a non-linear system of partial differential equations. It was proved in the
1990s that this system has global solutions on compact surfaces of any genus
g 0.
In this dissertation we study necessary and sufficient conditions for a surface to have CMC. We study the minimal case (characterized by mean curvature H 0) and the well-known Weierstrass representation for such surfaces.
Also CMC surfaces with rotational symmetry are considered and a generalization of the Weierstrass representation to surfaces of non-zero constant mean
curvature is presented. Finally we show that the only compact embedded
CMC surfaces in R 3 are spheres.
It has been my intention throughout this work to give references to the stated
results and credit to the work of others. The only part of this Master’s dissertation
which I claim is my own is the elementary proof of a special case of Ruh-Vilms
theorem for surfaces in R 3 given in Theorem 4.1.
ii
iii
Acknowledgements
I wish to thank my supervisor, Sigmundur Gudmundsson, for his time, knowledge and patience. In particular, I wish to express my gratitude for him inspiring me to study the wonderful subject of geometry.
Carl Johan Lejdfors
Contents
Short History
1
Chapter 1. Some basic surface theory
1.1. Notation
1.2. Isothermal coordinates
1.3. The tension field
3
3
5
7
Chapter 2. Minimal surfaces
2.1. Conformality of the Gauss map
2.2. The Weierstrass representation formula
9
9
10
Chapter 3. CMC surfaces of revolution
3.1. Kenmotsu’s solution
3.2. Delaunay’s construction
15
15
17
Chapter 4. CMC surfaces
4.1. Harmonicity of the Gauss map
4.2. Kenmotsu’s representation formula
23
23
24
Chapter 5. Compact CMC surfaces
Recent developments
33
35
Appendix A. Harmonic maps
37
Bibliography
41
v
Short History
In 1841 Delaunay characterized in [1] a class of surfaces in Euclidean space
which he described explicitly as surfaces of revolution of roulettes of the conics.
These surfaces are the catenoids, unduloids, nodoids and right circular cylinders.
Today they are known as the surfaces of Delaunay and are the first non-trivial
examples of surfaces having constant mean curvature, the sphere being the
trivial case.
In an appended note to Delaunay’s paper M. Sturm characterized these
surfaces variationally as the extremals of surfaces of rotation having fixed volume while maximizing lateral area. Using this characterization the following
theorem was obtained:
T HEOREM (Delaunay’s theorem). The complete immersed surfaces of revolution in R 3 having constant mean curvature are exactly those obtained by rotating
about their axis the roulettes of the conics.
These surfaces where also recognized by Plateau using soap film experiments. In 1853 J. H. Jellet showed in [2] that if is a compact star-shaped
surface in R 3 having constant mean curvature then it is the standard sphere. A
hundred years later, in 1956, H. Hopf conjectured that this, in fact, holds for
all compact immersions:
S
S
C ONJECTURE (Hopf ’s conjecture). Let be an immersion of an oriented,
compact hypersurface with constant mean curvature H 6= 0 in R n . Then must
be the standard embedded (n 1)-sphere.
S
Hopf proved the conjecture in [3] for the case of immersions of S 2 into R 3
having constant mean curvature and a few years later A. D. Alexandrov showed
the conjecture to hold for any embedded hypersurface in R n , see [4]. It was
widely believed that this conjecture was true until 1982 when Wu-Yi Hsiang
constructed a counterexample in R 4 . Two years later Wente constructed in [5]
an immersion of the torus T 2 in R 3 having constant mean curvature.
Wente’s construction has been thoroughly studied but has only been able
to create surfaces having genus g = 1. A different method for constructing surfaces in R 3 having constant mean curvature of any genus g 3 was
presented in 1987 by N. Kapouleas [6]. A proof of the fact that there exist
CMC-immersions of compact surfaces of any genus was published in [8] in
1995 by the same author.
1
CHAPTER 1
Some basic surface theory
In this chapter we introduce the notation to be used in this text. We also
introduce some basic results concerning isothermal coordinates and the tension
field of the Gauss map of a surface in R 3 .
1.1. Notation
S
D EFINITION 1.1. A non-empty subset
on R 3 is said to be a regular
there exists an open neighborhood U in
surface if for each point p 2
around p and a bijective map = (x ; y) : U ! R 2 such that its inverse
X : (U ) ! U
i. is a homeomorphism,
ii. is a differentiable map,
iii. (Xx Xy )(q) 6= 0 for all q 2 (U ).
The functions x ; y are called local coordinates around p. The map X is called a
local parametrization of around p.
S
f
f
S
S
f
S
S
S
S
be a regular surface in R 3 and p 2
be an arbitrary point. By
Let
a tangent vector to , at the point p, we mean the tangent vector 0 (0) of a
differentiable parametrized curve : ( ; ) ! with (0) = p. The set of
tangent vectors of at a point p 2 is called the tangent space of at p 2
and is denoted by Tp . A local parametrization X determines a basis
S
S
S
a
ee
S
Xx ; Xy
S
a
a
S
S
of Tp , called the basis associated with X .
On the tangent plane we have the usual induced metric from the ambient
space R 3 with the associated quadratic form Ip : Tp ! R called the first
fundamental form of at p 2 . Given a local parametrization X of and a
parametrized curve (t) = X (x(t); y(t)) for t 2 ( ; ) with p = (0) we have
the following form
S
a
S
ee
S
Ip ( 0 (0)) = Xx x 0 + Xy y0 ; Xx x 0 + Xy y0
a
= hXx ; Xx ip (x 0 )2 + 2 Xx ; Xy
= E(x 0 )2 + 2Fx 0 y0 + G(y0 )2 ;
p
x 0 y0 + Xy ; Xy
a
p
(y0 )2
S
(1.1)
where the values of E, F and G are computed for t = 0.
By condition (iii) of the definition of a regular surface (1.1) we have, given
a local parametrization X of a surface in R 3 at a point p 2 that the map
S
3
S
4
N:
1. SOME BASIC SURFACE THEORY
S ! S 2 defined by
N (p) = Xx Xy
(p)
Xx Xy (1.2)
S
is well defined. This map is known as the Gauss map of . The quadratic form
IIp defined in Tp by
IIp (v) =
dNp (v); v
is called the second fundamental form of at p. Given a local parametrization
X on at a point p 2 and, as above, letting be a parametrized curve such
that (t) = X (x(t); y(t)) for t 2 ( ; ) with p = (0) = X (x(0); y(0)), we get
S
S
a
S
S
a
ee
a
dNp (a0 ) = N 0 (x(t); y(t)) = Nx x 0 + Ny y0
Since hN ; N i = 1 we must have that Nx ; Ny
2 TpS and hence
Nx = a11 Xx + a21 Xy ;
Ny = a12 Xx + a22 Xy ;
for some functions aij . We find that
a
dNp ( 0 ); 0 =
IIp ( 0 ) =
(1.3)
a a
Nx x 0 + Ny y0 ; Xx x 0 + Xv x 0
= e(x 0 )2 + 2fx 0 y0 + g(y0 )2 ;
where
e=
f =
g=
hNx ; Xx i = hN ; Xxx i ;
Ny ; Xx = N ; Xxy = N ; Xyx =
(1.4)
Nx ; Xy ;
Ny ; Xy = N ; Xyy :
Using the terms from equations (1.1), (1.3) and (1.4) we arrive at
fF eG
gF fG
;
a12 =
;
2
EG F
EG F 2
eF fE
fF gE
a21 =
;
a22 =
;
2
EG F
EG F 2
known as the Weingarten equations.
Continuing by using that fXx ; Xy g is a basis for Tp and that N is orthogonal to both Xx and Xy we have that fXx ; Xy ; N g is a basis for R 3 . We find
that
1
2
Xxx = 11
Xx + 11
Xy + eN ;
a11 =
S
G
G
G
1
2
Xy + fN ;
Xyx = G12 Xx + G12
1
2
Xxy = G21
Xx + G21
Xy + fN ;
1
2
Xyy = G22
Xx + G22
Xy + gN :
(1.5)
The ijk are known as the Christoffel symbols and are invariant under isometries
(i.e. can be computed from the first fundamental form alone). Using that
(Xxx )y
(Xxy )x = 0;
1.2. ISOTHERMAL COORDINATES
5
(Xyy )x (Xxy )y = 0;
Nxy Nyx = 0;
we find that
G121 + f (G122 G111 )
1
2
gx = eG22
+ f (G22
G121 )
G112 ;
2
g G12
:
fx = e
ey
fy
g
(1.6)
These equalities are known as the Mainardi-Codazzi equations.
S
S
D EFINITION 1.2. Let be a surface in R 3 and p 2 an arbitrary point.
Let dNp : Tp ! Tp be the differential of the Gauss map. Then the determinant of dNp is called the Gaussian curvature K of at p. The negative half
of the trace of dNp is called the mean curvature H of at p.
S
S
S
S
In terms of the first and second fundamental forms K and H are given by
eg f 2
;
K =
EG F 2
eG 2fF + gE
H=
:
2(EG F 2 )
(1.7)
(1.8)
1.2. Isothermal coordinates
In this section we introduce the notion of isothermal coordinates which is
a useful tool in differential geometry.
S
S
be a surface in R 3 . Then local coordinates
D EFINITION 1.3. Let
(x ; y) : U ! R 2 on are said to be isothermal if there exists a strictly
positive function, called the dilation, : U ! R such that
S
E = hXx ; Xx i =
l2 =
l
S
Xy ; Xy = G ;
F = 0:
We have the following result regarding existence of isothermal coordinates
on an arbitrary surface in R 2 .
S
S
T HEOREM 1.4. Let be a differentiable surface in R 3 and p 2 be a point
on . Then there exists an open neighborhood U of p and isothermal coordinates
(x ; y) : U ! R 2 around p.
S
S
This was proved in the analytic case by Gauss. For a complete proof in the
general case please see [9]. Having chosen isothermal coordinates the mean
curvature simplifies
eG + gE 2fF
e+g
=
:
2
2(EG F )
2 2
The Christoffel symbols similarly simplify
H=
l
2
G111 = G122 = G221 = 21l2 lx ;
2
G112 = G121 = G222 = 21l2 ly :
(1.9)
6
1. SOME BASIC SURFACE THEORY
Using this we get the following form of the Mainardi-Codazzi equations (1.6):
ey
gx
G222 ;
1
fy = (e + g)G11
:
fx = (e + g)
The Weingarten relations (eq. 1.3) reduce to
e
f
a11 =
; a21 = a12 =
;
2
2
a22 =
(1.10)
g
(1.11)
l2 :
Suppose S is a surface in R 3 and f = (x ; y) : U S ! R 2 are local
isothermal coordinates on S . We may then consider the local coordinates
(x ; y) as a complex-valued map z = x + iy : U S ! C . The inverse
X : z(U ) C ! S can then be considered as map from an open subset z(U )
in C into S i.e. a local parametrization of S . We then have
l
l
1
Xx iXy :
2
The complex notation for surfaces in R 3 has many advantages which we will
be useful in chapters 2 and 3.
Letting h; i be the usual inner product in R 3 and let (; ) be the complex
bilinear extension of h; i in C 3 we have the following result.
Xz =
S
P ROPOSITION 1.5. Let be a surface in R 3 and let z = x + iy : U
C be local isothermal coordinates on . Then the inverse X : z(U ) C
z is conformal i.e. satisfies
S
4 (Xz ; Xz ) = jXx j2
4 (Xz ; Xz ) = jXx j2
2
Xy 2i Xx ; Xy = 0;
2
Xy + 2i Xx ; Xy = 0:
Conversely, if z = x + iy are local coordinates on
they are isothermal.
S !
! S of
(1.12)
S satisfying equation (1.12) then
P ROOF. The first statement follows by a direct computation. The reverse
implication follows by considering real and imaginary parts of equation (1.12).
S
S
f
P ROPOSITION 1.6. Let be a surface in R 3 and z = x + iy : U ! C
be local isothermal coordinates on with dilation . Then the inverse X : (U ) C ! of z satisfies
S
S
l
l
4Xz z = Xxx + Xyy = 2 2 HN ;
where N :
S ! S 2 is the Gauss map of S .
P ROOF. By a direct computation using the differentiated form of equation
(1.12) we have
2
4Xz z = 2 [(Xz z ; Xz ) Xz + (Xz z ; Xz ) Xz ] + 4 (Xz z ; N ) N
l
= Xxx + Xyy ; N N = 4(e + g)N
l
= 2 2 HN
1.3. THE TENSION FIELD
7
This immediately gives our sought result.
1.3. The tension field
In this section we give an explicit formula for the tension field (see appendix A) for maps from a surface in R 3 into S 2 in terms of local isothermal
coordinates.
S
f S
! S 2 be a map
P ROPOSITION 1.7. Let be a surface in R 3 and :
into the unit sphere S 2 in R 3 . If (x ; y) : U ! R 2 are local isothermal
coordinates on then the tension field ( ) of is locally given by
S
tf f
t(f) = l12 Df T
S
i.e. as the tangential part of the classical Laplacian
D = x2 + y2
2
2
in R 2 .
f S ! S2
P ROOF. By the definition of the tension field of a smooth map :
we have
t(f) =
2
X
re d f(ek )
k=1
f
d (rek ek ) ;
k
f
S
where r is the pull-back connection on the pull-back bundle 1 TS 2 over
via .
Let p 2 be an arbitrary point and (x ; y) : U ! R 2 be isothermal
coordinates around p. We then have
f
S
S
2
l
re ek = e ek + 2 2ek ( l12 )ek
2
l
1
= e ek +
2ek ( 2 )ek
2
l
k
g(ek ; ek ) grad
k
grad
k
1
1
l2
l2
for k = 1; 2. Then using the definition of the gradient grad we obtain
re e1 + re e2 = e e1 + e e2 + l
2
1
2
1
e1 (
2
1
1
l2 )e1 +e2( l2 )e2
grad
1 1
+ r1 l y
x
y
1 1 1 1 =
+
x
x
y
y
= r1
l x
l
l
l
f
1 d (re1 e1 + re2 e2 ) =
x
l
l2
l
l
This implies that
1
1
l
l
f
1
+
x
y
l
1
l
f
y
(1.13)
8
1. SOME BASIC SURFACE THEORY
The other term is given by
f T = 1 1 f + 1 2f T
l x l x
l x l x l x2
T
T
1 1 f
1 1 f 1 2f
re d f(e2 ) = l y l y = l y l y + l y2
re1 d f(e1 ) = 1 1
2
It follows by equations (1.13), (1.14) and (1.15) that
t(f) = re d f(e1) + re d f(e2) d f(re e1)
T
1 2f 2f
1 T
= 2
+
=
l x 2 y2
l2 Df
1
1
2
(1.14)
(1.15)
f
d (re2 e2 )
Harmonic maps generalize the concept of harmonic functions well known
from complex analysis. A harmonic map is one for which the tension field
vanishes everywhere and, as stated in Appendix A, arises as a critical point of a
certain variational problem.
S
f S
! S 2 be a map into
T HEOREM 1.8. Let be a surface in R 3 and :
2
3
the unit sphere S in R . If is conformal then it is harmonic.
S
f
P ROOF. Let p 2
be an arbitrary point and (x ; y) : U local isothermal coordinates around p. Then the conformality of
fx ; fy = 0
and
By differentiating we then obtain
fxx ; fy = fyx ; fx ;
hfxx ; fx i = fyx ; fy ;
and therefore
hfx ; fx i = fy ; fy
S ! R 2 be
f means that
:
f
f
yy ; fx =
xy ; fy ;
fyy ; fy = fxy; fx ;
f
f
xx + fyy ; fx = fyx ; fyy
yx ; fyy = 0;
fxx + fyy ; fy = fyx ; fxx + fxy ; fxx = 0:
These relations imply that
t(f) = l12 Df T = 0:
CHAPTER 2
Minimal surfaces
In this chapter we introduce some results concerning minimal surfaces. We
also prove the famous Weierstrass representation for minimal surfaces.
D EFINITION 2.1. A surface
curvature H satisfies H 0.
S in R 3 is said to be minimal if its mean
2.1. Conformality of the Gauss map
S
P ROPOSITION 2.2. Let be a minimal surface in R 3 . Then the Gauss map
N : ! S 2 of is conformal.
S
S
S
S
P ROOF. Let p 2 be an arbitrary point on and (x ; y) be local isothermal coordinates around p. Then it follows by
e+g
H=
=0
2 2
and equation (1.11) that
1
1
Nx = 2 eXx + fXy ;
Ny = 2 fXx eXy :
l
l
This implies that
Nx ; Ny = 0
l
hNx ; Nx i =
and
Ny ; Ny =
e2 + f 2
Hence N is conformal.
l2 0:
A partial reverse implication of the previous theorem is obtained via the
following.
S
S
P ROPOSITION 2.3. Let be a real analytic surface in R 3 and N : ! S 2
be a Gauss map of . If N is conformal then is either minimal or part of a
sphere.
S
S
P ROOF. For local isothermal coordinates (x ; y) on
0 = Nx ; Ny =
=
=
Let p 2
hood V
1
l2
l4
f
l
1
2
1
eXx + fXy ;
l2
S we have
fXx + gXy
ef hXx ; Xx i + fg Xy ; Xy
e + g = 2fH :
S be a point. Suppose H (p) 6= 0 then there exists an open neighbor S around p such that f jV = 0. For every point in q 2 V we have,
9
10
2. MINIMAL SURFACES
by equation (1.11), that Nx and Ny are
with Xx and Xy , respectively.
parallel
By conformality
we have that jNx j = Ny and, since (x ; y) are isothermal, that
jXx j = Xy . Hence q is umbilical i.e. the principal curvatures coincide. Let
k = k1 = k2 , where k1 and k2 are the principal curvatures. Differentiating
Nx = kXx and Ny = kXy gives
(kXx )y = (Nx )y = (Ny )x = (kXy )x
and since Xx and Xy are linearly independent we must have kx = ky = 0 so k
is constant. If k = 0 then H = 0 which contradicts the assumption. Hence
k 6= 0 and
N = kX + a
where a is a constant vector. Then X is a local parametrization for a sphere
having radius 1=k centered at a=k since
kX 1k ak2 = k 1k N k2 = k12 :
Thus by real analyticity is either minimal or part of a sphere.
S
2.2. The Weierstrass representation formula
The Weierstrass representation formula was first presented by Karl Weierstrass in [10]. It states that given two holomorphic functions defined on some
simply connected subset of C there exists an associated minimal surface. This
surface is unique up to motions.
S
f
S
T HEOREM 2.4. Let be a surface in R 3 and = x + iy : U ! C be
local isothermal coordinates on . Suppose U is an open simply connected subset
of . If X : z(U ) C ! is the inverse of z then is minimal if and only if
the derivative Xz : z(U ) ! C 3 is holomorphic.
S
S
S
S
P ROOF. This is a direct consequence of Proposition 1.6 and the fact that
a map f : U C ! C is holomorphic if and only if fz = 0.
Integration gives us the following corollary.
S
f
S
C OROLLARY 2.5. Let be a surface in R 3 and = x + iy : U ! C
be local isothermal coordinates on . Suppose U is an open simply connected subset
of . Then the inverse X : z(U ) C ! of z is given by
S
S
Z
X (z) = 2 Re
z
S
Xz (z)dz + C ;
(2.1)
z0
where C is some constant vector in R 3 .
P ROOF. We have
1
(Xx iXy )(dx + idy)
2
1
=
Xx dx + Xy dy + i(Xx dy
2
1
Xz d z =
Xx dx + Xy dy i(Xx dy
2
Xz dz =
Xy dx ;
Xy dx :
2.2. THE WEIERSTRASS REPRESENTATION FORMULA
Integrating dX = Xz dz + Xz d z = 2 Re Xz dz gives us our sought relation.
11
This corollary gives us the famous Weierstrass representation for minimal
surfaces.
T HEOREM 2.6 (Weierstrass Representation). Let V be an open simply connected subset of C . Suppose f : V ! C is holomorphic on V , g : V ! C is
meromorphic on V and the product fg 2 is holomorphic on V . Then X : V ! R 3
defined by
Z
X (z) = Re
z
Xz (z)dz ;
(2.2)
z0
where
Xz (z) = f (z)(1
is a minimal surface.
g(z)2 ; i(1 + g(z)2 ); 2g(z))
P ROOF. Using the above results the only thing we need to show is that
equation (2.2) define isothermal coordinates. This, however, follows by direct
computation using Proposition 1.5.
Examples of minimal surfaces are the surfaces of Sherk (Fig. 2.1) and Catalan (Fig. 2.2).
F IGURE 2.1. Sherk’s minimal surface. (f ; g) = ( 1 2z4 ; z)
An interesting observation is the fact that the Gauss map of minimal surfaces generated using Theorem 2.6 can be identified with the complex valued
function g.
P ROPOSITION 2.7. Let
strass representation
Z
X (z) = Re
S be a minimal surface in R3 given by the Weier-
z
f (z)(1
g(z)2 ; i(1 + g(z)2 ); 2g(z))dz :
z0
Then the Gauss map N :
projection , with g.
s
S!
S 2 of
S
may be identified, via stereographic
12
2. MINIMAL SURFACES
F IGURE 2.2. Catalan’s minimal surface. (f ; g) = i
P ROOF. Let G =
1
z
1
z3
;
z
s 1 Æ g then
(2 Re g ; 2 Im g ; g 2
1 + g2
Now the real and imaginary parts of
G=
f (1
1)
g 2 ); if (1 + g 2 ); 2fg
:
S in R 3 . Then
fg 1
i 1
fg g
g ; ( + g); 1 ; G =
g g + jg j2
2 g
2 g
2 g
and since jG j = 1 it is clear that G is a Gauss map for S .
represent two orthogonal tangent vectors on
1
=0
We can conclude that the above examples (Figures 2.1, 2.2) have bijective
Gauss map i.e. for every point p 2 S 2 there exists only one point on having
that point as a normal. Amongst the Enneper surfaces, defined by (f ; g) =
(1; z n ) for every n 2 N , only the case of n = 1 satisfies this property (see
Figures 2.3, 2.4 and 2.5).
S
F IGURE 2.3. First order Enneper surface. (f ; g) = (1; z)
2.2. THE WEIERSTRASS REPRESENTATION FORMULA
F IGURE 2.4. Second order Enneper surface. (f ; g) = (1; z 2 )
F IGURE 2.5. Third order Enneper surface. (f ; g) = (1; z 3 )
13
CHAPTER 3
CMC surfaces of revolution
In this chapter we study complete CMC surfaces with rotational symmetry. We present Kenmotsu’s modern solution given in [11] to the problem of
finding all such surfaces. Furthermore we describe the classical construction of
the same due to Delaunay, see [1].
S
D EFINITION 3.1. A surface in R 3 is said to have constant mean curvature (CMC) if and only if there exists a c 2 R such that H c.
3.1. Kenmotsu’s solution
f
f
Let : I R ! R with (s) = x(s); y(s) be a parametrization of some
regular planar C 2 curve. Assume that is an arclength parametrization and
that 0 is contained in the open interval I . Let be the surface of revolution
in R 3 defined by
2
f
S
(s; ) 7! (x(s); y(s) cos ; y(s) sin );
s
2 I ; 0 2p:
Then the first and second fundamental forms are given by
Ip = ds2 + y2 d 2 ;
IIp = (x 00 y0
Assuming y(s) > 0 for s
2Hy
x 0 y00 )ds2 + x 0 yd 2 :
2 I we have, by definition of H , that
x 0 x 00 yy0 + x 0 yy00 = 0;
s 2 I:
(3.1)
Multiplying by x 0 and y0 , respectively, and simplifying using the fact that
(x 0 )2 + (y0 )2 = 1 and
x 0 x 00 + y0 y00 = 0;
s
2 I;
we obtain
2Hyx 0 + (yy0 )0
1 = 0 and
2Hyy0
(yx 0 )0 = 0:
Setting Z (s) = y(s)y0 (s)+iy(s)x 0 (s) and combining these equations the following
first order complex linear differential equation is obtained
Z0
2iHZ
1 = 0;
s
2 I:
Restricting our attention to the case of H being constant we have:
If H = 0 then the solution is given by
Z (s) = s + C = s + c1 + ic2
15
(3.2)
16
3. CMC SURFACES OF REVOLUTION
for some C = c1 + ic2
2 C . This gives us
q
y(s) = jZ (s)j = (s + c1 )2 + c22;
x 0 (s) =
By integrating x we obtain
x = c2 arcsinh
(3.3)
Im Z
c2
=p
:
y
(s + c1 )2 + c22
s + c1
c2
hence
s + c1 = sinh
x
c2
c2 :
Substituting into equation (3.3) we obtain
y=
s
q
(s + c1 +
)2
c22
=
sinh
2
x
c2
c22
+
c22
= c2 cosh
x
c2
:
It is clear that this is a parametrization of a catenary.
If H 6= 0 then
1
1 e 2iHs + C e2iHs
Z (s) =
2iH
1
=
(1 + 2iHC) e 2iHs e2iHs
2iH
Bei(2Hs+) 1
=
;
2iH
(3.4)
where Bei = 1 + 2iHC for some B ; 2 R and C 2 C is an arbitrary
constant. Using the fact that y(s) > 0 we have by translation of the arclength
and by restricting our attention to H > 0
1 p
1 + B 2 + 2B sin 2Hs;
2H
Im Z
1 + B sin 2Hs
x 0 (s) =
=p
:
y
1 + B 2 + 2B sin 2Hs
y(s) = jZ j =
Hence the solution to equation (3.4) is the one-parameter family of surfaces of revolution having constant mean curvature H given by
f(s; H ; B) =
Z
0
s
1 + B sin 2Ht
dt ;
1 + B 2 + 2B sin 2Ht
1 p
2
1 + B + 2B sin 2Hs
(3.5)
2H
p
for any B 2 R and H > 0.
Studying for varying B (see Fig. 3.1) we find that (s; H ; 0) is a generating curve for a right circular cylinder and (s; H ; 1) is a generating curve for a
sequence of continuous half-circles centered on the x-axis. For 0 < B < 1 the
function x(s) increases monotonously whereas in the case of B > 1 it does not.
f
f
f
3.2. DELAUNAY’S CONSTRUCTION
17
T HEOREM 3.2 (Delaunay’s theorem). Any complete surface of revolution
with constant mean curvature is either a sphere, a catenoid or a surface whose
generating curve is given by (s; H ; B) for some B 2 R .
f
f
P ROOF. Let H 2 R be given and let H (s) be a generating curve parametrized
by arclength for a complete surface of revolution having constant mean curvature H . By uniqueness of solution of (3.2) we have H (s) = (s; H ; B) for
some B 2 R .
f
f
F IGURE 3.1. Solutions for H = 0:5 and B = 0; 0:5; 1; 1:5
3.2. Delaunay’s construction
The surfaces of Delaunay are constructed by rolling a conic ` along a
straight line in the plane and taking the trace of the focus F . This is called
a roulette of the conic `. This trace then describes a planar curve which is rotated about the axis along which it was rolled. This gives a surface of revolution
having constant mean curvature. The construction presented here is based on
the article [12] by J. Eells.
3.2.1. ` is a parabola. Let ` be a parabola given by ` : t 7! (t ; at 2 ) for
some a which we take to be strictly positive. Let F be the focus and A be
the vertex of ` (see Figure 3.2). Let K be a point on ` and denote by P the
intersection of the tangent line of ` at K with the horizontal axis. By solving
the line equation for the tangent at K we find that for K = `(t) = (t ; at 2 ) then
P = (t =2; 0). This implies that PK = OP. And since \FOP = \PKF we
K
F
A
P
O
F IGURE 3.2.
`
is parabola
18
3. CMC SURFACES OF REVOLUTION
F IGURE 3.3. Catenary
also have \OPF =
we have
\KPF = p2 . By definition of the trigonometric functions
FA = FP cos \AFP = FP cos \PFK :
Now let FP denote the x-axis along which our parabola rolls. Then the
ordinate of F in this system of coordinates is given by PF . Denote this by y.
We have
dx
cos \PFK = :
ds
where s is the arclength of the locus of F . This is equivalent to
a
dx
= cos ;
ds
a
(3.6)
where denotes the angle made by the tangent of F with the x-axis. We then
arrive at
c=y
dx
= yq
ds
dx
ds
dx 2
ds
or, equivalently,
dy
=
dx
+
r
y
=q
dy 2
1+
ds
y2
c2
c2
dy 2
dx
:
(3.7)
The solution to this differential equation is given by
y=
c x =c
e +e
2
x =c
= c cosh
x
c
which is a catenary (Fig. 3.3). The corresponding surface of revolution is the
catenoid (Fig. 3.4). The Gauss map of the locus of F into S 1 is given by x 7! x
where
a
cos
ax = dxds = yc
showing that the Gauss map is injective onto an open semicircle.
3.2. DELAUNAY’S CONSTRUCTION
19
F IGURE 3.4. Catenoid
3.2.2. ` is an ellipse. Let F and F 0 be foci of ` and O its center. Take
a point K on ` and let P and P 0 be the points on the tangent at K closest to
F and F 0 , respectively (Fig. 3.5). As above, letting PK be the x-axis and PF
(P 0 F 0 ) the ordinate y (y0 ). Let T and T 0 denote the intersection with the x-axis
of the tangent of the locus of F and F 0 , respectively.
We have \FKP = \F 0 KP 0 . Also the tangent of the locus of F (F 0 ) is
orthogonal to FK (FL0 ) and hence \KFT = \KF 0 T 0 = p2 . This gives us
y
dx
= sin \FKP = cos \FTP = ;
ds
FK
0
y
dx
= sin \F 0 KP 0 = cos \F 0 T 0 P 0 = :
0
ds
FK
From the characterization of the ellipse, FK + F 0 K = 2a for some a > 0, and
the pedal equation, PF P 0 F 0 = b2 for some b > 0, we find
y + y0 = 2a
dx
ds
and
yy0 = b2
T
P
F
K
O
F IGURE 3.5.
P·
F·
`
is ellipse
so
20
3. CMC SURFACES OF REVOLUTION
F IGURE 3.6. Undulary, H = 0:5, B = 0:5.
dx
+ b2 = 0:
ds
b we get the following cases (when the angle is obtuse and acute,
y2
Taking a respectively)
2ay
y2 2ay
dx
+ b2 = 0:
ds
A solution to this problem is given in Section 3.1
Z
x(s) =
y(s) =
(3.8)
p 1 +2B sin 2Ht dt ;
1 + B + 2B sin 2Ht
0
1 p
2
s
2H
q
(3.9)
1 + B + 2B sin 2Hs;
b
where H = 2a1 and B = 1 4H
The locus of either foci is called
2.
the undulary (Fig. 3.6). The corresponding surfaces is called the unduloid
(Fig. 3.7). The Gauss map of the undulary is given by x 7! x where
2
a
+b
ax = dxds = y 2ay
:
2
cos
2
F IGURE 3.7. Unduloid, H = 0:5, B = 0:5.
3.2. DELAUNAY’S CONSTRUCTION
21
3.2.3. ` is a hyperbola. We proceed as in the case of the ellipse but instead use the characterization FK
F 0 K = 2a > 0 of the hyperbola and
2
the pedal equation PF P 0 F 0 = b (Fig. 3.8). We arrive at the differential
equation
dx
y2 2ay
b2 = 0:
ds
K
P·
F
O
F·
P
F IGURE 3.8. Hyperbola
This differential equation can be solved in the same manner as for the ellipse
with the exception that B in equation (3.9) is given by
B=
r
b2
:
4H 2
Here the two loci fit together to form the curve known as the nodary (Fig. 3.9)
and the corresponding surface is called the nodoid (Fig. 3.10) The Gauss map
of the nodary is given by x 7! x where
1+
a
ax = y 2ayb :
2
cos
2
This map has no extreme points and is clearly surjective.
F IGURE 3.9. Nodary, H = 0:5, B = 1:5.
22
3. CMC SURFACES OF REVOLUTION
F IGURE 3.10. Nodoid, H = 0:5, B = 1:5.
CHAPTER 4
CMC surfaces
The main aim of this chapter is to give a new elementary proof of a special
case of the Ruh-Vilms’ theorem. We also present the Kenmotsu representation
formula for CMC surfaces with H 6= 0.
4.1. Harmonicity of the Gauss map
Let (M ; g) be an orientable m-dimensional Riemannian manifold, i : M !
be an isometric immersion and N : M ! Gpo (R m+p ) be the associated
R
Gauss map, mapping x 2 M to the oriented normal space of i(M) at i(x).
Then Ruh-Vilms’ theorem presented in [13] states that the tension field (N )
of N satisfies
(N ) = mrH ;
where rH is the covariant derivative of the mean curvature vector field H .
This implies that the Gauss map N is harmonic if and only if the mean curvature vector field H is parallel. For surfaces in R 3 this is equivalent to the
surface having constant mean curvature.
m+p
t
t
S
S
T HEOREM 4.1. Let be an oriented surface in R 3 . Then has constant
mean curvature if and only if its Gauss map N : ! S 2 is harmonic.
S
P ROOF. We prove that the following equation
t(N ) =
2 grad H
holds. Our sought result then follows trivially.
Let p 2 be an arbitrary point and (x ; y) : U
coordinates around p. Then we have
S
(Nxx
hNxx ; Xx i Xx +
)T =
=
1 l2
1 = 2 (
l
( ex
ex
S ! R 2 be isothermal
Nxx ; Xy Xy
l2
hNx ; Xxx i)Xx + (
hNx ; Xxx i)Xx + (
ey + (e + g)
G222
fx
Nx ; Xyx )Xy
This follows by using
ex =
fx =
hNx ; Xx i = hNxx ; Xx i + hNx ; Xxx i ;
x
x
Nx ; Xy = Nxx ; Xy + Nx ; Xyx ;
23
Nx ; Xyx )Xy :
24
4. CMC SURFACES
and the Mainardi-Codazzi equations (1.10). Similarly we have
(Nyy )T =
1 ( gx + (e + g)
2
l
G111
Ny ; Xxy )Xy + ( gy
Ny ; Xyy )Xy :
By using the Christoffel relations (1.9) we find
hNx ; Xxx i =
G111 Xx + G112 Xy 1
2
1
2
a11 G11
+ a21 G11
=
eG11
f G22
1
2
1
2
a11 G12
+ a21 G12
=
f G11
+ eG22
1
2
1
2
a12 G21
+ a22 G21
=
g G11
+ f G22
1
2
1
2
a12 G22
+ a22 G22
f G11
+ g G22
=
a11 Xx + a21 Xy ;
2
l
Nx ; Xyx = l2
Ny ; Xyx = l2
Ny ; Xyy = l2
=
Adding and using the above equations we get
hDN ; Xx i =
Nxx + Nyy ; Xx
1
(ex + gx ) + (e + g)
= 2
DN ; Xy
l
1
= 2
l
G111 hNx ; Xxx i
1
;
(ex + gx ) + 2(e + g)G11
l
Hence we have
t(N ) = l2 (DN )
= Nxx + Nyy ; Xy
1
= 2
(ey + gy ) + 2(e + g)
1
Ny ; Xxy
T
1
=
G222 :
ex + gx
e+g
l2 l2
+
1
+
ex + gx
l4
x
l4
y
e+g
l2 l2
1
=
ey + gy
l2 l2
+ (e + g)
l
+ (e + g)
1
2
ey + gy
l
2
l2 X
x
l2 X
y
x
y
(
(
1
l2 ) Xx
1
l2 )
Xy
1 1
1 1 1
=
(e + g) 2 + (e + g)
( ) Xx
x
x 2
1
1
1 1 1
+
(e + g) 2 + (e + g)
( ) Xy
y
y 2
= 2 grad H :
l
l
l
l l
l
l
l
l l
This proves our theorem.
4.2. Kenmotsu’s representation formula
In this section we show a corresponding result to Weierstrass’ representation formula for surfaces having non-zero constant mean curvature.
4.2. KENMOTSU’S REPRESENTATION FORMULA
25
Let S 2 be the unit sphere in R 3 . Cover S 2 by open sets Ui , i = 1; 2 where
U1 = S 2 fng and U2 = S 2 fsg and n and s are the north and south pole,
respectively. Let be the stereographic projection with respect to the north
pole n:
x1 + ix2
for x = (x1 ; x2 ; x3 ) 2 U1 :
(x) =
(4.1)
1 3
s
s
For a surface
ing composition
S in R 3 having Gauss map N : S ! S 2 consider the follow-
N
s
f: S !
S2 ! C
which we also call the Gauss map of S . This map is considered as a complex
mapping from a 1-dimensional complex manifold S in R 3 into the Riemann
sphere. Using this notation we have the following theorem presented in [14]
due to K. Kenmotsu.
T HEOREM 4.2 (Kenmotsu’s representation formula). Let V be an open
simply connected subset of C and H be an arbitrary non-zero real constant. Suppose : V ! C is a harmonic function into the Riemann sphere. If z 6= 0 then
X : V ! R 3 defined by
f
f
Z
z
X (z) = Re
Xz (z)dz ;
(4.2)
z0
with
Xz (z) =
for z
( 1)
1
H (1 + (z) (z))2
f f
f(z)2; i(1 + f(z)2); 2f(z) fz (z);
2 V , is a regular surface having f as a Gauss map and mean curvature H .
First we derive an explicit formula of the tension field of the Gauss map
of an arbitrary surface .
f: S ! C
S
S
f S
! C be a Gauss
P ROPOSITION 4.3. Let be a surface in R 3 and :
map on . If z = x + iy are local isothermal coordinates with dilation then
S
t(f) = l42
2
f
f f f
ff
2 :
1 + z z
z
z
l
(4.3)
f
P ROOF. Let z = x + iy be local isothermal coordinates and denote (z) =
u + iv. Then
4
g = 2 (dx 2 + dy2 ) and h =
(du2 + dv2 )
2
2
2
(1 + u + v )
which gives us Christoffel symbols on the Riemann sphere
2u
1
2
1
;
11 =
12 =
22 =
1 + u2 + v2
2v
2
1
2
:
11 =
12 =
22 =
2
1 + u + v2
l
G
G
G
G
G
G
26
4. CMC SURFACES
By the explicit formula for the tension field (eq. A.2) we get
t(u) = l2 Du + G
1
1
11
G
+
1
22
u
x
v
2
+
2
+
x
D
u
y
v
y
2 2 +2
G
1
12
u v
u v
+
x x
y y
;
t
where is the classical Laplacian. Adding this and the similar formula for (v)
we arrive at
l2t(f) = l2 t(u) + it(v)
= D(u + iv)
u iv
2
1 + u2 + v2
u
x
u
2
2i
2
2i
y
f f f
1 + ff
z z
2
f
f f f
=4
8
z
z
1 + ff
z z
=
Df
u v
x x
u v
y y
v
x
v
2
+
2 y
8
This proves our sought formula.
Next we show that a surface having prescribed mean curvature H satisfies
the following equation.
S
L EMMA 4.4. Let be a surface in R 3 having mean curvature H :
and let : ! C be a Gauss map of . Then
f S
S
H
where
y= l
1
g)=2
(e
2
S !R
f = y f ;
z
z
if .
P ROOF. We show that the following equation holds
f=
2 (X 1 + iX 2 )
H
1+ :
(4.4)
z
2
z
By direct computation using the Weingarten equations (1.11) together with
equation (4.1) we have
f=
z
z
Xx1
N 1 + iN 2
1 N3
+N
(1
3
N )
3
Xx1
Xy2
ff
=
N
Xx2
1
l
2 2 (1
Xx3
+
iXy2
1
N 3 )2
i
Xx2
+
iXx1
N
3
Xx2
+N
2
Xx3
+ (N + iN )
1
2
Xy3
e
+
iXx3
f
4.2. KENMOTSU’S REPRESENTATION FORMULA
+
Xy2
N
3
Xy2
+N
Using the definition
1 N = 2 Xx2 Xy3
2
Xy3
Xy1
N
Xx3 Xy2 ; Xx3 Xy1
l
3
+N
Xy1
1
27
Xy3
Xx1 Xy3 ; Xx1 Xy2
g :
Xx2 Xy1
(4.5)
and the fact that z = x + iy are isothermal together with equation
ff
(1 + )(1
N 3) = 2
our sought relation is obtained via
f=
1
U1 ;
on
(X 1 + iX 2 )
( 2)(e + g)
2 2 (1 N 3 )2
2H (X 1 + iX 2 )
=
(1 N 3 )2
z
H
(X 1 + iX 2 )
(1 + )2
=
:
2
z
By similar calculations we have
z
l
(4.6)
z
ff
f
y (1 + ff)2 (X 1 + iX 2) :
=
z
(4.7)
2
z
From equations (4.4) and (4.7) we may conclude that
H
f = y f :
z
z
The computations carried out in the proof of the previous lemma give us
a way of describing the dilation in terms of the Gauss map .
l
S
f
C OROLLARY 4.5. Let be an orientable surface in R 3 with mean curvature
! R and : ! C be the Gauss map of . Let z = x + iy be local
H:
isothermal coordinates on with dilation . Then
S
f S
S
l
S
= (1 + ) H ;
z 2
= (1 + )
:
z 2
f
l
f
l
ff j j
(4.8)
ff jyj
(4.9)
P ROOF. By equations (4.5) we have
2
l
4 Xz1 + iXz2 = 2 2 (1
2
4 Xz3 =
The above formulas gives us
N 3)
2
4 Xz3 ;
l2((N 1)2 + (N 2)2)2:
2
4 Xz1 + iXz2 =
l2(1
N 3 )2 :
(4.10)
From equations (4.4) and (4.6) we get our first sought result. Equation (4.9)
follows from equations (4.10) and (4.7).
28
4. CMC SURFACES
Next we show how the derivatives of the components of X are related to
the Gauss map.
S
P ROPOSITION 4.6. Let be an orientable surface in R 3 with mean curvature H : ! R and : ! C be the Gauss map of . Let X : V C !
be a local conformal immersion of , then the following equations hold on U1 :
S
f S
H
H
H
S
S
f f
ff
f f
ff
f f
ff
1 ( )2 ;
(1 + )2 z
1 + ( )2 =i
;
(1 + )2 z
= 2
:
2
(1 + ) z
X 1
=
z
X 2
z
X 3
z
s
S
(4.11)
P ROOF. Let ~ : U2 S 2 ! C denote the stereographic projection with
respect to the south pole s of S 2 and put = ~ Æ N . Then by similar calculations as in the proof of Lemma 4.4 we have
r s
fr
Using that
U1 \ U2 :
0=
(
r=
2 (X 1 iX 2 )
H
1+ :
(4.12)
z
2
z
= 1 we may conclude that the following equations hold on
fr) =
H
2
z
f r f r
rr
(X 1
f + r
iX 2 )
z
f
(X 1 + iX 2 )
+ +
z
f r
By ( + )=( + ) = 2 we have
H
(X 1
iX 2 )
z
f
+
2 (X
+ iX 2 )
= 0:
z
1
Rewriting as
f
H (1 + 2 )
X 1
= iH (1
z
and substituting into equation (4.4) we arrive at
H
X 2
z
=i
f2) Xz
2
f f
ff
1 + 2 :
(1 + )2 z
(4.13)
Similarly, we obtain the first formula of equation (4.11). The third equation
follows from
X 3 X 1
X 2
+i
= 2
:
z
z
z
(1 + )2
l
f
ff
This last equation follows by rewriting equation (4.6) to
4
f
l (1 + ff)2
z
z
X 3 X 1 + iX 2
=4
l2(1 N 3)(N 1 + iN 2):
z
z
X 3
X 1 + iX 2
2
4.2. KENMOTSU’S REPRESENTATION FORMULA
29
Expanding using equation (4.5) and separating out real and imaginary parts we
find that both vanish and the equation follows. If fz 6= 0 then by equations
(4.10) and (4.6) we have
H
X 3
z
1
2
l2 (1 +fff)2 X 1 +1 iX 2 = l2 (1 +fff)2 X1z + iX2z 2
jXz + iXz j
z
z
4f
Xz1 + iXz2
1
2
=
=
f
X
+
iX
:
z
z
(1 + ff
)2 (1 N 3)2
=
This implies the last equation of (4.11) by conjugating and substituting using
1
equation (4.13) and the similar formula for Xz . If fz = 0 then H must be
0.
P ROOF OF K ENMOTSU ’ S REPR . FORMULA . We have shown that for an
arbitrary surface having constant mean curvature H 6= 0 and complex Gauss
map : ! C the following equations hold
f S
S
f 2
f
ff
f f f
ff
2
jdz j2 ;
ds =
H (1 + ) z 2
2 = 0:
z
z 1 + z z
Similarly, given and H , we may construct a surface
parametrized by X using equation (4.2).
2
f
(4.14)
(4.15)
S which is locally
The last thing we need to show is that this representation is unique up to
conformal transformations.
S
f
T HEOREM 4.7. Let be a simply connected surface in R 3 and : U ! C
and e : U ! C be smooth mappings satisfying equation (4.15) for some constant
H 6= 0. We define a surface X and Xe by Theorem 4.2 i.e.
f
Z
X = Re
z
Xz dz
f
ff
z0
where
f
ff
f
ff
f
2
( 1) 1 1
1+ 2
Xz =
;
i
;
2
H
2 (1 + )2 (1 + )2 (1 + )2 z
Then the following conditions are equivalent:
i. There exists a holomorphic mapping = f (z) with f 0 (z) 6= 0 on and
a motion of R 3 such that Xe Æ f (z) = Æ x(z) for z 2 .
ii. There exists a holomorphic mapping = f (z) with f 0 (z) 6= 0 on
satisfying
w
S
w
f(z) = ef Æ f (z)
Xew
for
z
S
S
2 S:
P ROOF. Assume condition i. hold. We may assume = id. Then Xz =
Æ f 0 and Xz = Xew~ Æ f 0. We have
1
l
Xx + iXy =
f 0 1 e
e
X
+
i
X
jf 0 j ~ ~x ~y
l
30
4. CMC SURFACES
and therefore
2N (z) =
f
f
i
Xx + iXy
l2
Xx
e
iXy = 2N
Æ f (z):
Hence (z) = e Æ f (z) and ii. follows from equation (4.4).
Conversely suppose condition ii. hold. Then it follows from Theorem 4.2
e
X
= Xw Æ f 0 (z) that
z
X (z) = Xe (f (z)) + C ;
3
where C 2 R is some constant.
Kenmotsu’s representation formula allows us to, given a harmonic mapping : U C ! C , construct a surface having specified mean curvature
and as a Gauss map.
f
f
f
f
E XAMPLE 4.8. Let : C f0g ! C be given by (z) = 1=z and H =
1. The immersion X obtained via Theorem 4.2 is the standard immersion of
the unit sphere (see fig. 4.1) in R 3 :
1
X (z) =
z 2 C f0g:
(z + z ; i(z z ); zz 1) ;
1 + zz
F IGURE 4.1. Unit sphere in R3 .
f
f
E XAMPLE 4.9. Let : C
f0g ! C be given by (z) = 1=z2 . The by
Theorem 4.2 we have
Z 1 z 1 1 z 4
1 + z 4
z 2
X (z) = Re
;i
;2
dz :
H z0 2 1 + (zz ) 2 1 + (zz ) 2 1 + (zz ) 2
The corresponding surface of mean curvature H = 1 is given in figure 4.2.
4.2. KENMOTSU’S REPRESENTATION FORMULA
F IGURE 4.2. Surface with f = 1=z 2 with H = 1.
31
CHAPTER 5
Compact CMC surfaces
In this chapter we prove a result due to Alexandrov [4] stating that every
compact embedded CMC surface in R 3 is a sphere.
S
T HEOREM 5.1 (Alexandrov’s theorem). Let be a compact embedded surface in R 3 having constant mean curvature H 6= 0. Then is a sphere.
S
P ROPOSITION 5.2 (Willmore’s theorem). [15] Let S be a compact surface
3
embedded in R . Then
Z
with equality if and only if
S
H 2 dA 4
S
is a round sphere.
p
P ROOF. Let K + (p) = maxfK ; 0g then we will show that
Z
p
(5.1)
S
holds for any compact surface in R 3 . This follows from the observation that
the left hand side of the above equation represents the area of the image under
the Gauss map N of the part of S having K 0. So the only thing we need
to prove is that the image of N covers all of S 2 . Assume that N is oriented
outwards, then the point of S having minimal y-component has a normal
(0; 1; 0) and K 0 (since otherwise it would be a saddle point). But we
may orient our surface however we want (while maintaining K ) and hence we
have shown that there exists a point for every normal direction having K 0.
The surface area of the unit round sphere is 4p implying equation (5.1).
This gives us
Z
K + dA 4
jK j dA Z
p
K + dA 4 :
S
S
If K < 0 for any point then K < 0 in a neighborhood of that point and the
first inequality is strict. Hence equality implies K 0 everywhere and by the
previous equation and
H =
2
k1 + k2
2
2
= k1 k2 +
k1
k2
2
2
K
we have proved the first part of our result. If H 2 = K then k1 = k2 and we
know by Proposition 2.3 that is a round sphere. If is a round sphere then
H 2 = K proving our second statement.
S
S
33
34
5. COMPACT CMC SURFACES
S
L EMMA 5.3. [16] Let be a compact embedded surface in R 3 bounding a
domain D of volume V . If the mean curvature H of is positive everywhere, then
Z
1
dA 3V
(5.2)
SH
Equality holds if and only if is a round sphere.
S
S
P ROOF. Let N be the normal of
shell of defined by
S
S and for every e 2 [0; 1] let Se be the
Se = fp + eh(p)N j p 2 S g ;
where
h(p) = supfr j the point p is the unique nearest point on
S to the
point q at distance r from p along the normal S at pg:
S
The volume of e is then given by
Z
V =
where
Z
F =
h(p)
0
j(1
k1 t)(1
S
F dA;
k2 t)j dt =
(5.3)
Z
h(p)
1
2Ht + Kt 2 dt
0
Note that the definition of h we prevents overlap i.e. every point in the interior
of e lies on a unique normal to .
For every point q 2 D it lies in the closed shell e . Let d = dist(q; )
then the open ball Bq (d) centered at p with radius d satisfies Bq (d) \ = ;.
But there exists at least one point p 2 contained in the boundary of Bq (d).
For any q0 on the radius from q to p, if r is the distance from q0 to p then the
closed ball Bq (d) is contained in Bq (d) except for the point p. Hence p is the
unique point of having distance r from q0 . By definition of h(p) we must
have d h(p) and hence q lie on the closed shell.
Since all points of D are covered and the only points covered twice are in
the image of the boundary we have that the volume V of D is exactly equal to
the volume of the shell given by equation (5.3)
The equation
1
maxfk1(p); k2(p)g
(5.4)
h(p)
hold for every p 2 and we may conclude that
S
S
S
S
S
0
S
S
S
(1
k1 t);
(1
are non-negative for 0 t
h(p) and hence
h(p)
k2 t)j dt =
Z
F =
0
j(1
k1 t)(1
Z
k2 t)
h(p)
(1
k1 t)(1
k2 t) dt :
(5.5)
0
By the inequalities of geometric and arithmetic mean
(1
k1 t)(1
k2 t) (1
Ht)2
(5.6)
RECENT DEVELOPMENTS
35
and by equation (5.4) we have
1
h(p)
Then
Z
1=H
H:
1
3H
p
and by equation (5.3) we arrive at the sought equation (5.2).
Equality gives us, by Willmore’s theorem (5.2), that is in fact a round
sphere.
F
(1
Ht)2 dt =
S
P ROOF
OF
A LEXANDROV ’ S
ZZ
S
we have
ZZ
A=H
THEOREM .
hX ; N i dA =
hN ; X i dA = H
ZZZ
By the divergence formula
div X dx dy dz
D
ZZZ
div X dx dy dz = 3HV :
S
Then by Lemma 5.3 the surface must be a sphere since
ZZ
ZZ
1
1
A
dA =
dA =
= 3V :
H
H
H
Recent developments
Wente was the first to show that there exists a compact CMC immmersion
in R 3 which is not the standard sphere. He provided this by explicitly constructing an immersion of the torus T 2 in R 3 having constant mean curvature.
T HEOREM 5.4 (Wente’s counterexample). [5] There exists a countable number of isometrically distinct conformal immersion of T 2 into R 3 with constant
mean curvature H 6= 0.
The problem whether there exists immersions of genus g 2 was addressed by N. Kapouleas in [7, 8]. He successfully showed that there exists
CMC immersions of every genus g 2 by fusing Delaunay surfaces and
Wente tori.
T HEOREM 5.5. For any g 2 there exists infinitely many smooth CMCsurfaces immersed in R 3 having constant mean curvature H = 1 and genus g.
A recent development in the field is the DPW method [17, 18] named
after its creators J. Dorfmeister, F. Pedit and H. Wu. This method gives a
description of all immersed CMC surfaces in R 3 with or without umbilics. It
is often characterized as a Weierstrass type method as it allows the construction
of CMC-immersions from a meromorphic and a holomorphic function.
APPENDIX A
Harmonic maps
Harmonic mappings arise as critical points of a certain variational problem.
Let : (M ; g) ! (N ; h) be a smooth map between two Riemannian manifolds
of dimension m and n, respectively. We then define the energy density function,
e( ), of at p 2 M by
f
f
f
m
m
1
1X
1X
e(f) = traceg (f h) =
kd f(ei )k2h ;
h(d f(ei ); d f(ei )) =
2
2
2
i =1
i =1
where f h is the pull back of h via f and fei gmi=1 is a local orthonormal frame
of Tp M with respect to gp . Let p 2 M,
(x1 ; x2 ; : : : ; xm ) and
f
(y1 ; y2 ; : : : ; yn )
a
be local coordinates around p and (p) 2 N and for = 1; 2; : : : ; n put
a = y Æ . Then at each point q in a neighborhood of p
a
(
)
m
n
X
1 X ij
a
b
e( )(q) =
g (q)
hab ( (q))
(q)
(q) ;
2 i;j=1
xi
xj
a;b =1
f
f
f
f
f
f
where (g ij ) is the inverse of the matrix (gij ) with
gij = g(ei ; ej )
f on M by
e(f)dvolg ;
We define the energy or action integral of
f
Z
E( ) =
M
where volg = det(gij )dx1 dxn is the volume form on M.
A map F : ( ; ) M ! N is said to be a smooth variation of
satisfies
ee
(
f
F (0; p) = (p)
F: ( ; )M
ee
!N
f if it
p 2 M;
of class C 1 :
2 ( e; e) we define the smooth variation ft of f by
ft (p) = F (t ; p); p 2 M :
D EFINITION A.1. A map f : (M ; g) ! (N ; h) is said to be a harmonic
For every t
f
mapping if is a critical point to E at C 1 (M ; N ) i.e. for any smooth variation
< t < and 0 = we have
t : M ! N with
f
e
e
f f
d
E(ft ) = 0
dt
t =0
37
38
A. HARMONIC MAPS
Let
f
1
f
TN denote the pull-back of the tangent bundle TN of N via ,
f
1
TN = (p; u) 2 M
The variation vector field V along
d
dt
V (p) =
TN j p 2 M ; u 2 Tf(p)M
fdefined by
ft (p)
t =0
:
for all p 2 M ;
is a C 1 mapping from M into TN satisfying
V (p) 2 Tf(p) N
for all p 2 M :
We may then define a connection r called the pull-back connection, on
the set of smooth sections of X 2 1 TN by
f
dN 1
PfÆst V ( (t)) ;
d f(X ) V =
dt
t =0
s
(r V )(p) = N r
X
s
s
s
p 2 M;
s
where t 7! (t) 2 M is a C 1 curve in M satisfying (0) = p, 0 (0) = Xp 2
Tp M, and t is a curve given by t (s) = (s), 0 s t that is, the restriction
of to the part between p and (t). The map N PfÆst : Tf(p) N ! Tf(s(t)) N is
the parallel transport along a C 1 curve Æ t with respect to the Levi-Civita
connection N r on N .
s
s
s
s
f s
f
FACT A.2 (First variational formula). Let : (M ; g) ! (N ; h) be a smooth
map. For a smooth variation t of put
f f d
V (p) = ft (p)
dt
then
Thus
f
p2M
;
t =0
Z
d
E( t ) =
h(V ; ( ))dvolM
dt
M
t =0
is a harmonic mapping if and only if
f
t(f)(p) = 0
tf
for all p 2 M :
This is known as the Euler-Lagrange equation for harmonic maps. The
section ( ) of the pull-back bundle 1 TN is called the tension field of and
is given by
tf
f
t(f)(p) =
m
X
i=1
f
re d f(ek )
f
d (rek ek ) :
k
(A.1)
The tension field can be given in local coordinates by the following: let
(x1 ; x2 ; : : : ; xm ) and (y1 ; y2 ; : : : ; yn ) be local coordinates around p and (p) in
M and N , respectively. Then ( ) is given by
f
tf
t(f) =
t(f)g(p) =
m
X
i;j =1
g
ij
f
2 g
xi xj
n
X
g=1
m
X
k=1
t(f)g y
g
G
k
ij (p)
;
where
(A.2)
n
fg + X
fa fb
N g
G (f(p))
:
xk
a;b =1
ab
xi xj
A. HARMONIC MAPS
Here
tively.
39
Gijk , N Gabg are the Christoffel symbols on (M m; g) and (N n; h), respec-
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41
Master’s Theses in Mathematical Sciences 2003:E11
ISSN 1404-6342
LUNFMA-3023-2003
Mathematics
Centre for Mathematical Sciences
Lund University
Box 118, SE-221 00 Lund, Sweden
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