THE RIGIDITY OF THE SPHERE
April 25, 2016
A thesis submitted
To Kent State University in partial
Ful llment of the requirements for the
Degree of Master of Science
By
Paul C. Havens
April 2016
c Copyright
All rights reserved
Except for previously published materials Thesis Written By
Paul C. Havens
B.S., Kent State University, 2009
M.S., Kent State University, 2016
Approved by
Dmitry Ryabogin , Masters Advisor
Dr. Andrew Tonge , Chair, Departmentof Mathematical Sciences
Dr. James Blank , Dean, College of Arts and Sciences Contents
1 Introduction 1
2 Basic Dierential Geometry 4
2.1 First Fundamental Form ...... 4
2.2 Second Fundamental Form ...... 4
2.3 Curvatures ...... 5
2.4 Conformal Maps ...... 7
2.5 Gauss Map in Local Coordinates ...... 11
2.6 Christoel Symbols and the Mainardi-Codazzi Equations ...... 12
3 The Rigidity of the Sphere 15
3.1 Preliminaries ...... 15
3.2 Conclusion ...... 20
iii ACKNOWLEDGEMENTS
I would like to thank Dmitry Ryabogin for being my advisor and guiding me through the research process and working patiently with me. His knowledge and experience were of immense value to this writing.
I also extend my thanks to Benjamin Jaye and Artem Zvavitch for serving on my thesis comittee.
A special thanks goes to Carl Stitz for getting me started in my pursuit of my mathematics major and guiding me during my early undergraduate years.
I would like to thank my mother for her support throughout the process. Last but not least, I thank my friend Joy for her encouragement and support over the years.
iv 1 Introduction
Our thesis concerns the area of surfaces of constant mean curvatures. Such surfaces have been applied to the shapes of soap lms and soap bubbles. For a soap lm, we may imagine taking a bubble wand and dipping it into our bubble solution. The surface stretched across the opening to the bubble wand is an example of a constant mean curvature surface, particularly a surface of constant mean curvature zero. If we likewise take a wire frame of a polyhedron and dip it into our solution, the resulting shape is also a minimal surfaces. These surfaces have equal pressure on both sides of a sheet of soap lm, and have a mean curvature of zero. Such surfaces are called minimal surfaces, so called because surface tension minimizes the surface energy. We will in fact not be working with these directly, as we wish to be more general and study surfaces of constant mean curvature. For examples of these, we turn our eyes towards soap bubbles. Soap bubbles also have their surfaces minimized by surface tension, but they have a dierence in pressure between their interior and exterior. This translates to a constant yet nonzero mean curvature.
Beyond soap lms, surfaces of constant mean curvature have applications to the shape of a water droplet on a superhydrophobic surface as well as the design of air-supported structures such as various domes used in sports complexes.
We focus in this thesis on rigid surfaces. Intuitively, we are considering surfaces made of exible, inelastic materials. We call these surfaces rigid if it is not possible to deform them. A classical result of Cauchy says that a convex polyhedron in R3 that is made of rigid plates but exible hinges is a rigid surface. This thesis concerns an extension of this idea, a theorem due to Hopf: a regular surface with constant mean curvature which is homeomorphic to a sphere is a sphere.
We will show that a surface with this constant mean curvature is composed of so called umbilic points, which are exactly the points where the principle curvatures are equal. We will also show that such a surface must be either a sphere or a plane. Since we have assumed homeomorphism to a sphere, our surface must then be a sphere.
To proceed heuristically, we begin by introducing isothermal parameters, ds2 = E du2 + dv2, which not only are required conditions, but are nice to work with and simplify many of our calculations. We arrive at these from taking the coe cients of the rst fundamental form, E, F , and G, such that E = G and F = 0.
Among the rst items that we will need are the Gaussian curvature K and the mean curvature H. The Gaussian curvature is the product of the two principal curvatures, while the mean curvature is the average. Under our isothermal parameters, these take the forms
eg − f 2 K = = k k , E2 1 2 e + g k1 + k2 H = = . 2E 2
Where k1 and k2 are the principle curvatures.
1 The next major items that we will need are the Codazzi equations: ! e − g + f = EH , 2 v u u ! e − g − f = −EH . 2 u v v
2 To see how the two sets of items are related, we follow the example of Hopf and consider the function e − g φ(ζ) = φ(u, v) = − if. We have that 2 v u !2 u e − g t + f 2 |φ| 2 = , E E s e2 − 2eg + g2 + 4f 2 = , 4E2 s e2 + 2eg + g2 − 4eg + 4f 2 = , 4E2 s (e + g)2 eg − f 2 = − , 4E2 E2 v u !2 u k1 + k2 = t − k k . 2 1 2
Solving for zeros, we nd
v u !2 u k1 + k2 0 = t − k k , 2 1 2
2 |k1 − k2| 0 = . 2
|φ| Thus, = 0 when k = k . E 1 2 Chapter 2 is dedicated to the basics of dierential geometry that we will need for our purposes, including the de nitions and constructions of the rst and second fundamental forms, currvature, and the Codazzi equations mentioned above. We also prove the existence of isothermal parameters. A reader comfortable with these topics may move on to Chapter 3. In this chapter, we take the concepts we have gathered and develop our Codazzi equations into the form used above. We also introduce two parametrizations on the sphere which will work together to help us prove Hopf's theorem, which is handled in the nal section.
3 2 Basic Dierential Geometry
2.1 First Fundamental Form
De nition 1. [1] A subset S ⊂ R3 is a regular surface if for each p ∈ S, ∃ a nbd V in R3 and a map r : U → V T S of an open set U ⊂ R2 onto V T S ⊂ R3 such that
1. r is dierentiable; so r(u, v) = (x(u, v), y(u, v), z(u, v)), (u, v) ∈ U, then x(u, v), y(u, v), z(u, v) have continuous partials of all orders in U.
2. r is a homeomorphism. r is continuous by condition 1, so r has an inverse r−1 : V ∩ S → U which is continuous; i.e. r−1 is the restriction of a continuous map F : W ⊂ R3 → R2 de ned on an open set W containing V ∩ S.
2 3 3. (regularity) For each q ∈ U, the dierential drq : R → R is 1-1.
r is called a parametrization of coordinates in p. V ∩ S is called a coordinate neighborhood.
On each tangent plane Tp(S) of a regular surface S, we can get an inner product from the natural inner 3 product on R . We denote this as < w1, w2 >p, w1 · w2, and (when the meaning is clear) w1w2. De nition 2. Let R ⊂ S be a bounded region of a regular surface contained in the coordinate neighborhood of the parametrization 2 . The positive number RR , −1 , is r : U ⊂ R → S Q kru × rvk du dv = A(R) Q = r (R) the area of R.
2 De nition 3. [2] The quadratic form Ip(w) =< w, w >p= |w| ≥ 0 on Tp(S) is called the rst fundamental form of the regular surface S ⊂ R3 at p ∈ S.
In the basis {rurv} associated to a parametrization r(u, v) at p, since a tangent vector w ∈ Tp(S) is the tangent to the curve α(t) = r(u(t), v(t)), t ∈ (−, ), with p = α(0) = r(u0, v0), we get 0 0 0 Ip(α (0)) =< α (0), α (0) >p, 0 0 0 0 =< ruu + rvv , ruu + rvv >p, 0 2 0 0 0 2 = ruru(u ) + 2rurvu v + rvrv(v ) , = E(u0)2 + 2F (u0v0)2 + G(v0)2. (1)
2.2 Second Fundamental Form
ru × rv De nition 4. [3] Let N(q) = , q ∈ r(U). Let S ⊂ R3 be a normal surface. The map N : S → R3 |ru × rv| lies on the unit sphere S2 = (x, y, z) ∈ R3 : x2 + y2 + z2 = 1 . The map N : S → S2 is the Gauss Map of S.
Proposition 1. [4] The dierential dNp : Tp(S) → Tp(S) of the Gauss map is a self-adjoint linear map; that is, for an operator A, hAv, wi = hv, Awi for all u, v ∈ V .
4 Proof. dNp is linear, so it su ces to show hdNp (w1) , w2i = hw1, dNp (w2)i in the basis {w1, w2} of Tp(S). Let r(u, v) be a parametrization of S at p and {ru, rv} be the basis of Tp(S). If α(t) = r (u(t), v(t)) is a parametrized curve in S with α(0) = p, we have
0 0 0 dNp (α (0)) = dNp (ruu (0) + rvv (0)) , d = N (u(t), v(t)) | , dt t=0 0 0 = Nuu (0) + Nvv (0).
In particular, dNp(ru) = Nu and dNp(rv) = Nv. We need only show that hNu, rvi = hru,Nvi.
By taking the derivatives of hN, rui = 0 and hN, rvi = 0 relative to v and u respectively, we get
hNv, rui + hN, ruvi = 0,
hNu, rvi + hN, rvui = 0.
Therefore, hNu, rvi = − hN, ruvi =hNv, rui.
We now develop the second fundamental form.
De nition 5. The quadratic form IIp onTp(S), where IIp(v) = − hdNp(v), vi is called the second funda- mental form of S at p.
2.3 Curvatures
We now de ne our notion of curvature.
De nition 6. [5] Let C be a regular curve in S passing through p ∈ S, k the curvature of C at p, and cos θ = hn, Ni, where n is the normal vector to C and N is the normal vector to S at p. The number kn = k cos θ is then called the normal curvature of C ⊂ S at p.
For each p ∈ S, there exists an orthonormal basis {e1, e2} of Tp(S) such that dNp(e1) = −k1e1, dNp(e2) = −k2e2. Moreover, k1 and k2 (k1 ≥ k2) are the maximum and minimum of the second fundamental form IIp restricted to the unit circle in Tp (S).
We consider a regular curve C ⊂ S parametrized by α(s), where s is the arc length of C, and α(0) = p. Denote by N(s) the restriction of the normal vector N to the curve α(s), we get hN(s), α0(s)i = 0. Thus,
hN(s), α00(s)i = − hN 0(s), α0(s)i , 0 0 0 IIp (α (0)) = − hdNp (α (0)) , α (0)i , = − hN 0(0), α0(0)i , = hN(0), α00(o)i ,
= hN, kni (p) = kn(p). (2)
De nition 7. [6] The maximum normal curvature and minimum normal curvature are k1 and k2, respectively. The corresponding directions, given by the eigenvectors e1 and e2, are called the principle directions.
5 Let v ∈ Tp(s), |v| = 1, and let e1and e2 form an orthonormal basis of Tp(S). Then v = e1 cos θ + e2 sin θ, where θ is the angle from e1 to v in the orientation Tp(S). The normal curvature Kn along v is given by
Kn = IIp(v) = − hdNp(v), vi ,
= − hdNp (e1 cos θ + e2 sin θ) , e1 cos θ + e2 sin θi ,
= he1k1 cos θ + e2k2 sin θ, e1 cos θ + e2 sin θi , 2 2 = k1 cos θ + k2 sin θ. (3)
Consider a linear map A : V → V of a two dimensional vector space with basis {v1, v2} of V . We have that
det A = a11a22 − a12a21,
trA = a11 + a22,
−k1 0 where A = (aij) in the basis {v1, v,2}. Here, since dN = , det dN = (−k1)(−k2) = k1k2 and 0 −k2 trdN = −(k1 + k2). This leads us to a de nition.
De nition 8. [7] Let p ∈ S and let dNp : Tp(S) → Tp(s) be the dierential of the gauss map. det dNp = K, the Gaussian Curvature of S at p. trdNp = −2H, where H is the mean curvature of S at p.
In terms of the principal curvatures, we write:
K = k1k2 k1 + k2 H = (4) 2
Our next de nition will provide the nal key to the proof of Hopf's theorem.
De nition 9. [8] If at p ∈ S, we have k1 = k2, then p is called an umbilical point of S. In particular, the planar points k1 = k2 = 0 are umbilical points. Proposition 2. [8] If all points of a connected surface S are umbilical points, then S is either contained in a sphere or a plane.
Proof. Let p ∈ S and x (u, v) be a parametrization of S at p such that the coordinate neighborhood V is connected.
Suppose q ∈ V is an umbilic point; then for any vector w = a1ru + a2rv in Tq(S),
dN(w) = λ(q)w, (5) where λ = λ(q) is a real dierentiable function in V .
First, we show that λ(q) is constant in V . To this end, the above equation becomes
Nua1 + Nva2 = λ (rua1 + rva2) and thus
Nu = λru
Nv = λrv
6 Dierentiating the rst equation in v and the second in u, we obtain
λurv − λvru = 0
Since ru and rv are linearly independent, we conclude that λu = λv = 0 for all q ∈ V . Since V is connected, λ is constant in V .
Now, if , and thus constant in . Thus, . λ ≡ 0 Nu = Nv = 0 N = No = V hr(u, v),N0iu = hr(u, v),N0iv = 0 Thus, hr(u, v),N0i = constant, and all points r(u, v) of V belong to a plane.
If λ 6= 0, then the point r(u, v) − (1/λ) N(u, v) = c(u, v) is xed, since ! ! ! ! 1 1 r(u, v) − N(u, v) = r(u, v) − N(u, v) = 0 λ λ u v 1 Since |r(u, v) − c|2 = , all points of V are contained in a sphere of center q and radius 1/ |λ|. Thus, in a λ2 neighborhood of p ∈ S, our Proposition holds.
Since S is connected, given another pont q ∈ S, there exists a continuous α : [0, 1] → S with α(0) = p, α(1) = q. For each point of α(t) ∈ S, there exists a neighborhood Vt in S contained in either a sphere or a −1 S −1 plane with α (Vt) being an open interval in [0, 1]. We have [0, 1] ⊂ α (Vt). Since [0, 1] is a closed t∈[0,1] −1 interval, it is covered by a nite number of elements from α (Vt) by the Heine-Borel Theorem. Thus, α[0, 1] is covered by a nite number of neighborhoods of Vt.
If the points of one of these neighborhoods are on a plane, then so are all other neighborhoods, and since r is arbitrary, all points of S belong to this plane.
Similarly, if the points of a neighborhood is on the sphere, all points of S belong to the sphere.
2.4 Conformal Maps
De nition 10. [9] A dieomorphism ϕ : S1 → S2 is called a conformal map if for all p ∈ S1 and all v1, v2 ∈ Tp (S1) we have
2 hdϕp (v1) , dϕp (v2)i = λ (p) hv1, v2ip
2 where λ is a nowhere-zero dierential function on S1. The surfaces S1 and S2 are called conformal.
It will soon be useful for us to de ne isothermal coordinates.
2 De nition 11. The parameters u and v of a regular surface r are called isothermal if Ip (w) = µ (z, z) dzdz = λ2 (u, v) du2 + dv2 = Edu2 + 2F dudv + Gdv2, where z is complex and z denotes the complex conjugate. Essentially, we have that E = G and F = 0. Theorem 1. [9] A change of coordinates preserves isothermal parameters if and only if the coordinate change is either complex analytic or a composition of a complex analytic coordinate change with complex conjugation.
7 Proof. Let ds2 = λ (u, v) du2 + dv2 = µ (z, z¯) dzdz¯. Let z = z (w ¯) be a complex analytic change of ! ! ∂z dz dz coordinates such that = 0. Then dz = dw and dz = dw. We then have ∂w¯ dw dw
dz dz ds2 = µ (z, z) dzdz = µ z (w, w) , z (w, w) dwdw dw dw
which is still isothermal in form, as is the complex conjugate.
Assume now that isothermal parameters are preserved, z = z (w) = z (w, w), and that z is neither analytic ∂z ∂z nor a composite of an analytic function with complex conjugation, so that and are simultaneously not ∂w ∂w ! ∂z ∂z ∂z ∂z identically zero. Thus, there are points where 6= 0 and where 6= 0. Since dz = dw + dw , ∂w ∂w ∂w ∂w we have
2 ∂z ∂z 2 2 2 ds = µ dw + dw = µ Edu + 2F dudv + Gdv ∂w ∂w
∂z ∂z Since there are points where niether nor are zero, we cannot conclude that F = 0, and thus isothermal ∂w ∂w parameters are not preserved. Theorem 2. [10] Assume E, F, and G are real valued analytic functions. There exists a parametrization 2 2 2 2 2 2 for which Ip (w) = ds = µ (u, v) du + dv = Edp + 2F dpdq + Gdq .
Proof. Factoring, we nd that √ ! √ ! √ F + i EG − F 2 √ F − i EG − F 2 Edp2 + 2F dpdq + Gdq2 = Edp + √ dq Edp + √ dq E E
We wish to nd new coordinates u = u (p, q) and v = v (p, q) to complete our theorem. To that end, we wish to nd a complex valued function λ = λ (p, q) such that √ ! √ F + i EG − F 2 λ Edp + √ dq = du + idv, E √ ! √ F − i EG − F 2 λ Edp + √ dq = du − idv. E
Multiplying, we nd
λ2ds2 = du2 + dv2, ds2 = |λ|−2 du2 + dv2 .
8 √ ! √ F + i EG − F 2 It is su cient to nd functions u, v, and λ satisfying λ Edp + √ dq = du + idv = E ! ! du dv du dv + i dp + + i dq. We break this down into dp dp dq dq
√ du dv λ E = + i , dp dp √ ! F + i EG − F 2 du dv λ √ = + i . E dq dq
Thus, we have ! 1 du dv λ = √ + i , E dp dp √ ! F + i EG − F 2 du dv λ √ = + i , E dq dq √ ! ! E du dv λ = √ + i . F + i EG − F 2 dq dq
Equating the lambdas, we achieve ! √ ! ! 1 du dv E du dv √ + i = √ + i , E dp dp F + i EG − F 2 dq dq ! ! p du dv du dv F + i EG − F 2 + i = E + i , dp dp dq dq ! du p dv p du dv du dv F − EG − F 2 + i EG − F 2 + F = E + Ei , dp dp dp dp dq dq
√ F − i EG − F 2 And by multiplying by , we get E √ ! √ ! F − i EG − F 2 p du dv F − i EG − F 2 du dv F + i EG − F 2 + i = E + i , E dp dp E dq dq
2 2 ! ! F + EG − F du dv p du dv + i = F − i EG − F 2 + i , E dp dp dq dq ! du dv du p dv dv p du G + iG = F + EG − F 2 + i F − EG − F 2 , dp dp dq dq dq dq
Which break down further into
du p dv du F − EG − F 2 = E , dp dp dq ! p du dv dv EG − F 2 + F = E , dp dp dq
9 and
du du p dv G = F + EG − F 2 , dp dq dq dv dv p du G = F − EG − F 2 . dp dq dq
Solving, we nd du du F − E ∂v dp dq = √ , ∂p EG − F 2 du du G − F ∂v dp dq = √ , ∂q EG − F 2 dv dv E − F ∂u dq dp = √ , ∂p EG − F 2 dv dv F − G ∂u dq dp = √ . ∂q EG − F 2
∂2u ∂2u ∂2v ∂2v With continuous second partials of u and v, we have = and = . We de ne a dierential ∂p∂q ∂q∂p ∂p∂q ∂q∂p operator L de ned as d d d d F − E F − G ∂ dp dq ∂ dq dp L = √ + √ , ∂q EG − F 2 ∂p EG − F 2 where Lu = 0 and Lv = 0. Lµ = 0 is Beltrami's equation. It is a fact from partial dierential equations that if E, F , and G are real analytic, then Beltrami's equation has solutions u and v with the map (p, q) → (u (p, q) , v (p, q)) being one to one. A proof of the existence of solutions to Beltrami's equation can be found in [11]. If we choose u as such a solution, then du du F − E ∂v dp dq = √ , ∂p EG − F 2 du du G − F ∂v dp dq = √ ∂q EG − F 2 determine u, and λ is determined by the equations √ ! √ F + i EG − F 2 λ Edp + √ dq = du + idv, E √ ! √ F − i EG − F 2 λ Edp + √ dq = du − idv. E
10 2.5 Gauss Map in Local Coordinates
This following section is heavily based on the formulations from DoCarmo. [12]
ru × rv Let N = , where r(u, v) is a parametrization at a point p ∈ S of a surface S, and let α(t) = ||ru × rv|| r(u(t), v(t)) be a parametrized curve on S, with α = p. Assume that all further functions in this section take their values at the point p.
0 0 0 The tangent vector to α(0) at p is α = ruu + rvv and 0 0 0 0 dN(α ) = N (u(t), v(t)) = Nuu + Nvv .
Since Nu and Nv belong to Tp(S),
Nu = a11ru + a21rv,
Nv = a12ru + a22rv, (6) therefore
0 0 0 0 0 dN(α ) = (a11u + a12v )ru + (a21u + a22v )xv, and thus 0 0 u a11 a12 u dN 0 = 0 . v a21 a22 v
The expression of the second fundamental form in the basis {xu, xv} is given by 0 0 0 0 0 IIp(v) = − hdNp(α), α i = − hNuu + Nvv , ruu + rvv i , = e(u0)2 + 2fy0v0 + g(v0)2,
where, since hN, rui =hN, rvi = 0,
e = − hNu, rui = hN, ruui ,
f = − hNv, rui = hN, ruvi = hN, rvui = − hNu, rvi ,
g = − hNv, rvi = hN, rvvi .
By (1),
−f = hNu, rvi = a11F + a21G,
−f = hNv, rui = a12E + a22F,
−e = hNu, rui = a11E + a21F,
−g = hNv, rvi = a12F + a22G. (7)
Where E, F , and G are the coe cients of the rst fundamental form in the basis {ru, rv} by (1) In matrix form, we may express (7) as
e f a a EF − = 11 21 , (8) f g a12 a22 FG −1 e f EF a a − = 11 21 , f g FG a12 a22 1 e f G −F a a − = 11 21 . EG − F 2 f g −FE a12 a22
11 For ease of calculation and later necessity, we assume that we are uner an isothermal paramatrization, where E = G, F = 0. Under this, we obtain:
1 e f E 0 a a − = 11 21 (9) E2 f g 0 E a12 a22
Under this parametrization, we optain the aij:
− e a = , 11 E − f a = , 12 E − f a = , 21 E − g a = . (10) 22 E
eg − f 2 From (9), we obtain the gaussian curvature K = det(a ) = = k k and the mean curvature ij E2 1 2 e + g H = = k + k 2E 1 2 a11 + k a12 k1and k2 are roots of det = 0 a21 a22 + k
2.6 Christoel Symbols and the Mainardi-Codazzi Equations
We continue in our isothermal parametrization (E = G, F = 0). We once again use DoCarmo as our basis for this section.[13]
De nition 12. We may nd the Christoel Symbols by taking the derivatives of xu, xv,, and N in the basis (xu, xv,N), we obtain:
u v xuu = Γuuxu + Γuuxv + eN, u v xuv = Γuvxu + Γuvxv + fN, u v xvu = Γvuxu + Γvuxv + fN, u v xvv = Γvvxu + Γvvxv + gN,
Nu = a11xu + a21xv,
Nv = a12xu + a22xv. (11)
Thus Nu and Nv are
12 − e − f N = x + x , u E u E v − f − g N = x + x . v E u E v
We take the inner product of the relations from (11) with xu and xv to obtain (once again in our isothermal parametrization):
1 E = hx x i = Γu E + Γv F = Γu E, 2 u u, uu uu uu uu 1 − E = hx x i = Γu F + Γv G = Γv E, 2 v v, uu uu uu uu 1 E = hx x i = Γu E + Γv F = Γu E, 2 v u, uv uv uv uv 1 E = hx x i = Γu F + Γv G = Γv E, 2 u v, uv uv uv uv 1 − E = hx x i = Γu E + Γv F = Γu E, 2 u u, vv vv vv vv 1 E = hx x i = Γu F + Γv G = Γv E. (12) 2 v v, vv vv vv vv
We solve, using (11)
(xuu)v = (xuv)u (13)
u v u v u v u v Γuuxuv+Γuuxvv+eNv+(Γuu)vxu+(Γuu)vxv+evN = Γuvxuu+Γuvxvu+fNu+(Γuv)uxu+(Γuv)uxv+fuN, (14)
− f − g Γu (Γu x +Γv x +fN)+Γv (Γu x +Γv x +gN)+e( x + x )+(Γu ) x +(Γv ) x +e N = uu uv u uv v uu vv u vv v E u E v uu v u uu v v v − e − f Γu (Γu x + Γv x + eN) + Γv (Γu x + Γv x + fN) + f( x + x ) + (Γu ) x + (Γv ) x + f N. uv uu u uu v uv vu u vu v E u E v uv u u uv u v u (15)
We then gather the coe cients of xu, xv, and N.
! ! − ef − eg Γu Γu + Γv Γu + (Γu ) + x + Γu Γv + Γv Γv + (Γv ) + x +(Γu f + Γv g + e ) N = uu uv uu vv uu v E u uu uv uu vv uu v E v uu uu v ! − fe − f 2 Γu Γu + Γv Γu + (Γu ) + x +(Γu Γv +Γv Γv +(Γv ) + )x +(Γu e+Γv f +f )N, uv uu uv vu uv u E u uv uu uv vu uv u E v uv uv u (16)
13 So by equating coe cients of xu, xv, and N, respectively:
! ! − ef − fe Γu Γu + Γv Γu + (Γu ) + − Γu Γu + Γv Γu + (Γu ) + = 0, uu uv uu vv uu v E uv uu uv vu uv u E ! − eg − f 2 Γu Γv + Γv Γv + (Γv ) + − (Γu Γv + Γv Γv + (Γv ) + ) = 0, uu uv uu vv uu v E uv uu uv vu uv u E u v u v (17) (Γuuf + Γuug + ev) − (Γuve + Γuvf + fu) = 0,
− ef + fe Γv Γu − Γv Γu + (Γu ) − (Γu ) + = 0, uu vv uv vu uu v uv u E − eg + f 2 Γv Γv − Γv Γv + (Γv ) − (Γv ) + = 0, uu vv uv vu uu v uv u E u v u v (18) Γuuf + Γuug + ev − Γuve − Γuvf − fu = 0,
v u v u u u −ΓuuΓvv + ΓuvΓvu − (Γuu)v + (Γuv)u = FK = 0, eg − f 2 Γv Γv − Γv Γv + (Γv ) − (Γv ) = EK = . (19) uu vv uv vu uu v uv u E
From this process, and applying again to (xvv)u = (xvu)v, we arrive at the Mainardi-Codazzi equations by equating coe cients of N: De nition 13. The Mainardi-Coddazi equations are as follows:
u v u v (20) ev − fu = Γuve + Γuvf − Γuuf − Γuug,
u v u v (21) fv − gu = Γvve + Γvvf − Γuvf − Γuvg.
14 3 The Rigidity of the Sphere
3.1 Preliminaries
Theorem 3. (Hopf) A regular surface with constant mean curvature which is homeomorphic to a sphere is a sphere.
From now on, we let U ⊂ R3 be an open connected subset of R2 and let x : U → S be an isothermal parametrization (i.e. E = G, F = 0) of a regular surface S. Identify R3 with the complex plane C by setting u + iv = ζ, (u, v) ∈ R2, ζ ∈ C. ζ is called the complex parameter corresponding to x. Let φ : x(U) → C e − g be the complex valued function given by φ(ζ) = φ(u, v) = − if = φ + iφ , where e, f, and g are the 2 1 2 coe cients of the second fundamental form of S. Proposition 3. The Mainardi-Codazzi equations can be written in the isothermal parametrization x as ! ! e − g e − g + f = EH , − f = −EH . 2 v u 2 u v u v
e + g Proof. Recall that H = . We compute the following: 2E
2E(eu + gu) − (e + g)2Eu EH = , u 4E eu + gu (e + g)Eu = − . 2 2E
By (12), we have:
Eu = Γu = Γv = −Γu . 2E uu uv vv
It then follows that
u eu + gu (e + g)Γ EH = − uu. u 2 1
15 Proof. In fact, we may distribute and rewrite our christoel symbols further:
eu − gu + 2gu EH = + eΓu − gΓv , u 2 vv uv eu − gu = + g + eΓu − gΓv , 2 u vv uv eu − gu = + f , (22) 2 v
Since u v v u (as v u ). gu + Γvve − Γuvg = fv + Γvvf − Γuvf = fv Γvv = Γuv
Very similar calculations yield the following:
2E(ev + gv) − (e + g)2Ev −EH = −E · , v 4E2 − ev − gv (e + g)Ev = + , 2 2E
And by using
Ev = −Γv = Γu = Γv , 2E uu uv vv we nd:
v − 2ev − gv + ev (e + g)Γ −EH = + vv, v 2 1 ev − gv = − e + eΓu − gΓv , 2 v uv uu ev − gv = − f . (23) 2 u
Since v u u v . −fu = −fu − Γuvf + Γuuf = −ev + Γuve − Γuug Corollary 1. The mean curvature H of x(U) ⊂ S is constant if and only if φ is an analytic function of ζ (i.e. (φ1)u = (φ2)v = −(φ2)u.
ev − gv eu − gu Proof. (⇒) If H is constant, H = H = 0, thus equations (22) and (23) give = f , = −f , u v 2 u 2 v so (φ1)u = (φ2)v and (φ1)v = −(φ2)u.
ev − gv eu − gu (⇐) f is analytic, so (φ ) = (φ ) and (φ ) = −(φ ) , = f , = −f , and H = H = 0. 1 u 2 v 1 v 2 u 2 u 2 v u v ! ∂ 1 ∂ ∂ De nition 14. The complex derivative operator is de ned as = − i . ∂ζ 2 ∂u ∂v
16 ! ∂x ∂y ∂z Proposition 4. φ(ζ) = −2 hx ,N i, where x = , , . ζ ζ ζ ∂ζ ∂ζ ∂ζ
Proof. First, we observe that 0 =< xζ , N >ζ =< xζζ , N > + < xζ ,Nζ >, since xζ is in the tangent plane and N is the normal vector.
And so < xζ ,Nζ >= − < xζζ , N >.
1 1 We also have x = (x − ix ), x = (x − x − 2ix ), and thus ζ 2 u v ζζ 4 uu vv uv
1 e − g −2 < x ,N >= 2 < x , N >= (x · N − x · N − 2ix · N) = − if. ζ ζ ζζ 2 uu vv uv 2 Proposition 5. Let f : U ⊂ C → V ⊂ C be a one-to-one complex function given by f(ζ) = f(u + iv) = x(u, v)+iy(u, v) = η. Then (x, y) are isothermal parameters on a regular surface S if and only if f is analytic and f 0(ζ) 6= 0, ζ ∈ U.
Proof. That f is conformal is identical to theorem 1 and theorem ??, which is further identical to f being analytic and f 0 (ζ) 6= 0. Proposition 6. Let y = x ◦ f −1 be the corresponding parametrization from above, and de ne ψ(η) = −2 hyη,Nηi. On x(U) ∩ y(V ), we have
!2 dη φ(ζ) = ψ(η) (24) dζ
Proof. The proof is as follows:
!2 !2 dη dη ψ(η) = −2 hy , N i , dζ η η dζ * + dη dη = −2 x ◦ f −1(η) , N f −1(η) , η dζ η dζ * + dη dη = −2 x (ζ(η)) , N (ζ(η)) , η dζ η dζ * + ∂x(ζ(η))dη ∂N ∂ζ dη = −2 , , ∂η dζ ∂ζ ∂η dζ * + ∂x ∂ζ dη ∂N = −2 , , ∂ζ ∂η dζ ∂ζ * + ∂x ∂N = −2 , , ∂ζ ∂ζ = φ (ζ) .
Lemma 1. We may cover the sphere minus a point S2 −(0, 0, 1) ⊂ R3 by the stereographic projection η from the north pole given by the following properties:
17 1. η(N) = 0 1 2. η−1 = (2Re(η), 2Im(η), ηη¯ − 1) 1 + ηη¯ x + iy 3. η(x, y, z) = 1 − z
Proof. We begin with the stereographic projections on the sphere x2 + y2 + z2 = 1
We begin with our rst projection, N = (0, 0, 1), which takes (x, y, z) −→ (u, v, 0). Under this, x = u(1 − z), y = v(1 − z). We then have
(u2 + y2)(1 − z)2 + z2 = 1 z2 − 1 = −(u2 + y2)(1 − z)2, 1 − z2 = (u2 + y2), (1 − z)2 − (1 − z) + 2 1 + z = , 1 − z 1 − z = (u2 + y2), 2 = 1 + u2 + y2, 1 − z 2 1 − z = , 1 + u2 + v2 − 1 + u2 + v2 z = . 1 + u2 + v2
1 x + iy Therefore, η−1 = (2Re(η), 2Im(η), ηη¯ − 1), η(x, y, z) = 1 + ηη¯ 1 − z
Lemma 2. We may cover the sphere minus a point S2 − (0, 0, −1) ⊂ R3 by the stereographic projection ζ from the south pole given by the following properties:
1. ζ(S) = 0 1 2. ζ−1 = 2Re(ζ), −2Im(ζ), 1 − ζζ¯ 1 + ζζ¯ x − iy 3. ζ(x, y, z) = 1 + z
Proof. Our second projection follows similarly. Let S = (0, 0, −1), (x, y, z) −→ (u, v, 0), and x = u(1 + z), y = v(1 + z).
18 (u2 + y2)(1 + z)2 + z2 = 1, z2 − 1 = −(u2 + y2)(1 + z)2, 1 − z2 = (u2 + y2), (1 + z)2 − (1 + z) + 2 1 − z = , 1 + z 1 + z = (u2 + y2), 2 = 1 + u2 + y2, 1 + z 2 1 + z = , 1 + u2 + v2 2 1 + z = , 1 + u2 + v2 1 − u2 − v2 z = . 1 + u2 + v2
1 x − iy Therefore, ζ−1 = 2Re(ζ), −2Im(ζ), 1 − ζζ¯, ζ(x, y, z) = . 1 + ζζ¯ 1 + z Lemma 3. η and ζ cover S2, and η = ζ−1.
Proof. It is clear that η and ζ cover S2.
(x − iy)(x + iy) x2 + y2 These parameters intersect on W = S2\{(0, 0, 1), (0, 0, −1)}, and ηζ = = = (1 − z)(1 + z) 1 − z2 x2 + y2 = 1. Therefore, η = ζ−1 on the sphere minus the two poles. x2 + y2 Proposition 7. Assume ∃ on each coordinate neighborhood ζ and η analytic functions φ(ζ), ψ(η), where η and ζ are as above, such that (24) holds in W. Then φ ≡ 0 (hence ψ(η) ≡ 0).
!2 dη Proof. By (24), φ(ζ) = ψ(η) . To use Liouville's Theorem, we need only show that these functions dζ are bounded.
On U ∩ V (i. e. the sphere without the poles), we have that these functions are bounded, as they are from the sphere to the sphere. To take care of the poles:
!2 dη φ : x(U) → , ψ : y(V ) → . On U ∩ V , φ(ζ) = ψ(η) , C C dζ
!2 d φ(ζ(0, 0, 1)) = φ(∞) = ψ(η(0, 0, 1)) η(0, 0, 1) = ψ(0) · 0 = 0, dζ
!2 d φ(ζ(0, 0, −1)) = φ(0) = ψ(η(0, 0, −1)) η(0, 0, −1) = ψ(0) · 0 = 0, dζ
19 !2 d ψ(η(0, 0, −1)) = ψ(∞) = φ(ζ(0, 0, −1)) ζ(0, 0, −1) = φ(0) · 0 = 0. dη
Thus, at the poles, our functions are equal to 0, and are, in fact, bounded. This allows us to use Liouville's Theorem, which tells us not only that our functions constant, but they are in fact identically 0.
3.2 Conclusion
Proof. (Of Theorem 3) Let S ⊂ R3 be a regular surface with constant mean curvature homeomorphic to the sphere.
S is compact. There exists a conformal dieomorphism ϕ : S → S2 of S onto the unit sphere S2 (by the uniformization theorem for Riemann surfaces). Let ζ˜ and η˜ be the complex parameters corresponding under 2 ˜ ˜ ˜ ϕ to parameters ζ and η of S given in Lemmas 1 and 2. We will also let E˜, F˜, G˜, e˜, f, g˜, k1, k2 be the coe cients of the rst and second fundamental forms and the principle curvatures, respectively, on S.
! e˜ − g˜ By Proposition 3 and Corollary 1, φ(ζ˜) = − if˜ is analytic since S has constant mean curvature. 2 The similar function ψ(˜η) is also analytic for the same reason. By Proposition 6, they are related by
!2 dη˜ φ(ζ˜) = ψ(˜η) dζ˜
e˜ − g˜ By Lemma 7, φ(ζe) ≡ 0. Thus, − if˜ = 0, so f˜ = 0, and e˜ =g ˜. From the beginning, we found that 2 e˜ +g ˜ ˜ ˜ ˜ e˜ g˜ ˜ H = = k1 + k2, so k1 = = = k2, so all points of S are umbilical points, and S must be either 2E˜ 2E˜ 2E˜ a plane or a sphere by Proposition 2. Since S is homeomorphic to a sphere, S is, in fact, a sphere.
20 References
[1] DoCarmo, Mandfredo P., Dierential Geometry of Curves and Surfaces, Prentice-Hall Inc., Upper Saddle River, New Jersey, 1976, pp. 52 [2] DoCarmo, Mandfredo P., Dierential Geometry of Curves and Surfaces, Prentice-Hall Inc., Upper Saddle River, New Jersey, 1976, pp. 92 [3] DoCarmo, Mandfredo P., Dierential Geometry of Curves and Surfaces, Prentice-Hall Inc., Upper Saddle River, New Jersey, 1976, pp. 136
[4] DoCarmo, Mandfredo P., Dierential Geometry of Curves and Surfaces, Prentice-Hall Inc., Upper Saddle River, New Jersey, 1976, pp. 140 [5] DoCarmo, Mandfredo P., Dierential Geometry of Curves and Surfaces, Prentice-Hall Inc., Upper Saddle River, New Jersey, 1976, pp. 145
[6] DoCarmo, Mandfredo P., Dierential Geometry of Curves and Surfaces, Prentice-Hall Inc., Upper Saddle River, New Jersey, 1976, pp. 144 [7] DoCarmo, Mandfredo P., Dierential Geometry of Curves and Surfaces, Prentice-Hall Inc., Upper Saddle River, New Jersey, 1976, pp. 146
[8] DoCarmo, Mandfredo P., Dierential Geometry of Curves and Surfaces, Prentice-Hall Inc., Upper Saddle River, New Jersey, 1976, pp. 147 [9] Dubrovin, B.A., Fomenko, A.T., Novikov, S.P., Modern Geometry-Methods and Applications Part 1. The Geometry of Surfaces, Transformation Groups, and Fields, Springer Science+Business Media, New York, 1984, pp. 108-109
[10] Dubrovin, B.A., Fomenko, A.T., Novikov, S.P., Modern Geometry-Methods and Applications Part 1. The Geometry of Surfaces, Transformation Groups, and Fields, Springer Science+Business Media, New York, 1984, pp. 110-111 [11] L. Bers, Riemann Surfaces, New York University, Institute of Mathematical Sciences, New York, 1957- 1958, pp. 19-35.
[12] DoCarmo, Mandfredo P., Dierential Geometry of Curves and Surfaces, Prentice-Hall Inc., Upper Saddle River, New Jersey, 1976, pp. 153-157 [13] DoCarmo, Mandfredo P., Dierential Geometry of Curves and Surfaces, Prentice-Hall Inc., Upper Saddle River, New Jersey, 1976, pp. 232-235
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