ALEKSANDROV S THEOREM: CLOSED SURFACES WITH CONSTANT MEAN CURVATURE

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1 ALEKSANDROV S THEOREM: CLOSED SURFACES WITH CONSTANT MEAN CURVATURE ALAN CHANG Abstract. We present Aleksandrov s proof that the only connected, closed, n- dimensional C 2 hypersurfaces (in R n+1 ) of constant mean curvature are the spheres. Contents 1. Introduction Reducing the problem to planes of symmetry 2 2. Mean curvature The mean curvature of a graph CMC surfaces and the isoperimetric inequality 4 3. Elliptic second-order PDE Positive-definiteness of (ã ij ) An elliptic PDE for the difference of two graphs Local uniqueness results 8 4. Aleksandrov s method of moving planes First case: an interior point in common Second case: a boundary point in common The Gaussian curvature case Gaussian curvature of graphs Surfaces of constant Gaussian curvature 14 Appendix A. The Hopf lemma and the strong maximum principle 16 Appendix B. Nonlinear second-order PDE 16 Acknowledgments 18 References 18 Date: June 10,

2 ALEKSANDROV S THEOREM: CLOSED SURFACES WITH CONSTANT MEAN CURVATURE 2 1. Introduction Recall that a closed surface is one that is compact and without boundary. Aleksandrov proved that if a closed, connected C 2 surface has constant mean curvature, then the surface is a sphere. In this paper, we present his proof. In Section 2, we present the concept of mean curvature. In Section 3, we derive some relevant consequences of the Hopf lemma and the strong maximum principle, which come from the theory of second-order elliptic PDE. The heart of this paper is Section 4. We present the method of moving planes, which Aleksandrov developed to show that any connected, closed, CMC ( constant mean curvature ) surface must have a plane of symmetry in every direction. By Lemma 1.4, it follows that such a surface is a sphere. The idea is geometric and quite intuitive. (Turn to Section 4 to see all the pictures!) In Section 5, we sketch an argument that closed, connected C 2 surfaces of constant Gaussian curvature are spheres as well. The setup (i.e., the analogues of Section 2 and Section 3) requires a few modifications from the CMC case. Once the setup is complete, the method of moving planes itself can be applied with no changes at all. Remark 1.1. In order for mean curvature (and Gaussian curvature) to be defined, the surface must be C 2. For the rest of this paper, the term surface always refers to a C 2 n-dimensional hypersurface embedded in R n+1. Remark 1.2. The method of moving planes has subsequently been used with great success to resolve many other questions (e.g., [BN91, GNN79, Ser71]). Usually, such questions deal with symmetry of solutions to certain PDE. Remark 1.3. Aleksandrov s original paper is in Russian, but an English translation is available [Ale62]. In addition, H. Hopf has written an exposition on Aleksandrov s theorem [Hop83, Chapter VII] Reducing the problem to planes of symmetry. Suppose we have a connected closed surface S, and suppose we want to show it is a sphere. By the following lemma, it suffices to show that S has a plane of symmetry in every direction. Lemma 1.4. If S R n+1 is a connected, closed surface with a plane of symmetry in every direction, then S is a sphere. Proof. Translate S so that the center of mass is at the origin. Then every plane of symmetry of S must contain the origin. Since S has a plane of symmetry in every direction, every plane containing the origin is a plane of symmetry.

3 ALEKSANDROV S THEOREM: CLOSED SURFACES WITH CONSTANT MEAN CURVATURE 3 Every element of the orthogonal group O(n+1) can be written as a finite composition of reflections through planes containing the origin. Since S is invariant under each reflection, it is invariant under any orthogonal transformation. Since S is connected and closed, it must be a sphere. 2. Mean curvature 2.1. The mean curvature of a graph. Let U R n be open and u C 2 (U). Then the graph of u gives a surface S = {(x, u(x)) R n R : x U}. (2.1) (When we say the surface u, we are referring to the graph of u.) Since S is a level set of the map (x, x n+1 ) x n+1 u(x), the downward unit normal vector ν(x) at the point (x, u(x)) is given by the gradient of this map. Hence, ν(x) = ( u(x), 1) 1 + u 2. (2.2) For any x U, the point (2.2) lies in the lower unit half-sphere, which we can parametrize as {(y, 1 y 2 ) : y R n, y < 1}. (2.3) We take x = (x 1,..., x n ) as in (2.1) to be coordinates for S, and we take y = (y 1,..., y n ) as in (2.3) to be coordinates for the lower unit half-sphere. In these coordinates, the Gauss map of S is given by x y = u(x) 1 + u(x) 2. (2.4) The second fundamental form of the surface is the matrix ( ) n yi II(x) =, (2.5) x j i.e., the Jacobian of the Gauss map (2.4). By definition, the principal curvatures of a surface at x are the eigenvalues of II(x), and the mean curvature H(x) at x is the average of the principal curvatures at x. Thus, H(x) = 1 tr II(x). This gives (with dependence on x suppressed) n ( ) H = 1 n div u (2.6) 1 + u 2 ( ) = 1 u n D2 u( u, u), (2.7) 1 + u 2 (1 + u 2 ) 3/2

4 ALEKSANDROV S THEOREM: CLOSED SURFACES WITH CONSTANT MEAN CURVATURE 4 where D 2 u( u, u) = ( ij u)( i u)( j u). (2.8) (That is, we view the n n matrix D 2 u as a bilinear form R n R n R.) Thus, we can write H = ã ij ( u) ij u, (2.9) where ( ) ã ij (p) = 1 δ ij n p i p j. (2.10) 1 + p 2 (1 + p 2 ) 3/2 Remark 2.1. The mean curvature depends on a choice of orientation for the surface. If we had considered the opposite orientation instead (i.e., with the upward unit normal vector), the sign of the mean curvature would be flipped. Our choice of orientation is so that the upper unit hemisphere u(x) = 1 x 2 has mean curvature H(x) = CMC surfaces and the isoperimetric inequality. Remark 2.2. This section is not needed for Aleksandrov s theorem; it is here to provide some intuition for CMC surfaces. Let U R n be open and let u C 2 (U) C(Ū). Define the surface area and volume operators by A[u] = 1 + u 2 dx (2.11) U V[u] = u dx. (2.12) U 2 Lemma 2.3. Let u C (U) C(Ū), let f C( U), and let λ R. Then u minimizes A[u] subject to the constraints V[u] = λ (2.13) u = f on U if and only if u is a CMC surface satisfying (2.13). Proof. We define two sets of functions: F = {u C 2 (U) C(Ū) : V[v] = λ and u = f on U} (2.14) C c,0(u) = {v C (U) : V[v] = 0 and v has compact support in U}. (2.15)

5 ALEKSANDROV S THEOREM: CLOSED SURFACES WITH CONSTANT MEAN CURVATURE 5 The optimization problem of interest is to minimize A[u] subject to the constraint u F. For u F and v Cc,0(U), we have u + tv F for all t R. We compute the derivative d A[u + tv] dt = t=0 U u v dx. (2.16) 1 + u 2 Let MC(u) : U R denote the mean curvature of u. Using the fact that v vanishes on U and recalling (2.6), integration by parts gives us ( ) d A[u + tv] u dt = div v dx = n MC(u)v dx. (2.17) t=0 1 + u 2 U Since A is a convex functional and F is a convex set, there is at most one critical point u, and such a point (if it exists) must be the global minimum of A. From (2.17), it follows that u is a critical point (hence minimum) if and only if MC(u)v dx = 0 for all v Cc,0(U). (2.18) U Thus, we have reduced the problem to showing that u has constant mean curvature if and only if (2.18) holds. One direction is straightforward: if u has constant mean curvature, then for all v Cc,0(U), MC(u)v dx = MC(u) v dx = 0. U U Conversely, suppose (2.18) holds. We can take v Cc,0(U) to be of the following form: let φ C (R n ) have support in a ball {x R n : x < r} for some r. Let y, z U. If r is sufficiently small then v(x) := φ(x y) φ(x z) has support in U and hence v Cc,0(U). (The picture is this: to get from u to u + v, first, we apply a small perturbation φ to u around some point y U. In order to keep the volume under the graph unchanged, we then apply the opposite perturbation φ around another point z U.) From (2.18), we have MC(u)(x + y)φ(x) dx = MC(u)(x + z)φ(x) dx. (2.19) { x <r} { x <r} It follows that MC(u)(y) = MC(u)(z), so the mean curvature of the graph must be constant. This proves the converse direction. Corollary 2.4. Fix λ > 0. Suppose that among all open connected C 2 regions in R n+1 of volume λ, X is the one with the least surface area. Then X is a CMC surface. Proof. Suppose for contradiction that S := X is not a CMC surface. Then there is some point x S around which the mean curvature is not constant. Near x, S is a graph if we choose the coordinate system appropriately. By Lemma 2.3, we can alter U

6 ALEKSANDROV S THEOREM: CLOSED SURFACES WITH CONSTANT MEAN CURVATURE 6 the surface locally (near x) to decrease the surface area without changing the volume underneath the graph. Remark 2.5. Of course, we know by the isoperimetric inequality that X in Corollary 2.4 must be a ball. Thus, this corollary is just saying that spheres have constant mean curvature, which is obvious. (Aleksandrov s theorem gives the converse: the only CMC surfaces are spheres.) The reason we present this corollary is to highlight an important point in its proof: in the proof, we had to consider a local neighborhood of S which could be expressed as a graph of a function. This is because Lemma 2.3 only applies to graphs. In particular, we cannot directly apply Lemma 2.3 to closed surfaces. While it is true that every surface can be written locally as a graph, CMC only guarantees minimal surface area for each of these local charts. This is not the same as a global minimum, since the total surface area of a CMC surface could potentially be decreased by a global perturbation, i.e., one that requires simultaneous changes in multiple charts. As a result, we may not deduce Aleksandrov s theorem from Lemma 2.3. Remark 2.6. To help understand this distinction between local and global minimization, let us consider a different minimization problem: suppose we want to find a closed path in R 3 of minimal length subject to the constraint that the path lies in S 2 (the unit sphere). Clearly, the global minimum is achieved by constant paths. However, consider a curve along a great circle. Since a great circle is a geodesic of S 2, it is impossible to decrease the length by modifying the curve locally. In this case, locally means in an open set of S 2 that is a graph in R 3. There is an obvious global modification that decreases the length: just shift the whole curve away from the great circle so that the curve becomes a smaller circle. Remark 2.7. It is in fact possible to define the second fundamental form II of a n- dimensional surface S embedded in an (n + 1)-dimensional Riemannian manifold M other than R n+1. For more on this, see, e.g., [Lee97, Chapter 8]. With the generalized definition of curvature, it is still true that constant mean curvature implies locally minimal. Returning to the example in Remark 2.6, if we take M to be the Riemannian manifold S 2 and S M to be a great circle, then S has mean curvature zero (as a submanifold of M). Thus, for some manifolds M, it is possible to find locally minimal surfaces that are not globally minimal. 3. Elliptic second-order PDE 3.1. Positive-definiteness of (ã ij ). Recall the coefficients ã ij given in (2.10). Let Ã(p) = (ã ij (p)) be the n n matrix. We view à as a symmetric bilinear form on Rn,

7 ALEKSANDROV S THEOREM: CLOSED SURFACES WITH CONSTANT MEAN CURVATURE 7 so we can write (Ã(p))(ξ, ξ) for ξ Rn. We will suppress dependence on p and just write Ã(ξ, ξ). The following lemma shows that à is positive-definite. Furthermore, the positivedefiniteness is uniform if p is restricted to a bounded subset of R n : Lemma 3.1. Let V R n be bounded. Then there exists an absolute constant λ > 0 such that Ã(ξ, ξ) λ ξ 2 for all p V and ξ R n. (3.1) Proof. Using (2.10), we compute ( ) Ã(ξ, ξ) = ã ij ξ i ξ j = 1 ξ 2 (p ξ)2. (3.2) n 1 + p 2 (1 + p 2 ) 3/2 Applying Cauchy-Schwarz ((p ξ) 2 p 2 ξ 2 ), we have ( ) Ã(ξ, ξ) 1 ξ 2 n p 2 ξ 2 = 1 ξ 2. (3.3) 1 + p 2 (1 + p 2 ) 3/2 n (1 + p 2 ) 3/2 Since p ranges in the bounded set V, the quantity (1 + p 2 ) 3/2 is bounded above An elliptic PDE for the difference of two graphs. Lemma 3.2. Let u, v C 2 (U) C(Ū) and suppose u, v, D2 u, D 2 v are bounded on U. Suppose u and v are CMC surfaces with the same mean curvature. Then w := u v solves an SMP-admissible PDE. (See Definition A.1.) Proof. From (2.9), we have H = H = ã ij ( u) ij u (3.4) ã ij ( v) ij v. (3.5) While H depended on x in (2.9), here, H is a constant. Subtracting these two equations gives us [ã ij ( u) ij u ã ij ( v) ij v] = 0. (3.6) Let w = u v. We can eliminate ij u from (3.6), giving us [ã ij ( u) ij w + (ã ij ( u) ã ij ( v)) ij v] = 0. (3.7)

8 ALEKSANDROV S THEOREM: CLOSED SURFACES WITH CONSTANT MEAN CURVATURE 8 Using the fundamental theorem of calculus, we have [ 1 ] ã ij ( u) ã ij ( v) = ã ij ((1 t) u + t v) dt w, (3.8) where the gradient ã ij is taken with respect to p. (Recall (2.10).) Setting we have 0 a ij (x) = ã ij ( u(x)) (3.9) 1 b k (x) = ij v(x) k ã ij ((1 t) u(x) + t v(x)) dt, (3.10) a ij (x) ij w(x) + 0 b k (x) k w(x) = 0 on U. (3.11) k=1 Since u, v, D 2 u, D 2 v are bounded on U, the functions a ij and b i are bounded on U. Combining the boundedness of u with Lemma 3.1, (a ij ) is uniformly elliptic. Remark 3.3. In (3.11), the coefficients a ij and b k depend on u and v. Thus, this PDE is not very useful if u, v, w are unknowns and we want to solve for w. In our situation, we do not start with a PDE and try to find its solution. We are actually doing the opposite: we start with a solution w and in Lemma 3.2, we find a PDE that w satisfies. The lemma is useful because the PDE turns out to be of a certain form (SMP-admissible). In general, the types of computation in Lemma 3.2 are useful for uniqueness proofs (as in our case see Section 3.3) and for obtaining estimates. Remark 3.4. In Lemma 3.2 as well as the two lemmas that follow in Section 3.3, we make the assumption that u, v, D 2 u, D 2 v are bounded. This is an unimportant technical assumption; in practice, we can choose our region U so that these derivatives are bounded. (That is what we do in Section 4.) Remark 3.5. Lemma B.2, used in Section 5, is a generalization of Lemma Local uniqueness results. We present two results that give us sufficient conditions to conclude that two CMC surfaces are the same. Lemma 3.6 (Uniqueness via interior condition). Let U = {x R n : x < r}, (3.12) i.e., U is the open ball of radius r in R n. Suppose u, v C 2 (U) C(Ū) are CMC surfaces with the same mean curvature. Suppose u, v, D 2 u, D 2 v are bounded on U. If u v on U and u(0) = v(0), then u = v on U.

9 ALEKSANDROV S THEOREM: CLOSED SURFACES WITH CONSTANT MEAN CURVATURE 9 Proof. Let w = u v. Then w 0 on U and w attains its maximum at 0. Because w solves an SMP-admissible PDE (Lemma 3.2), we may apply the strong maximum principle (Lemma A.4) to w. Thus, w must be constant, hence identically zero. Lemma 3.7 (Uniqueness via boundary condition). Let U = {x R n : x < r and x n > 0}, (3.13) i.e., U is the open upper half-ball of radius r in R n. Suppose u, v C 2 (U) C(Ū) are CMC surfaces with the same mean curvature. Suppose u, v, D 2 u, D 2 v are bounded on U. If u v on U, u(0) = v(0), and n u(0) = n v(0), then u = v on U. Proof. Let w = u v. Then w 0 on U and w attains its maximum at 0, a boundary point of U. Because of Lemma 3.2, we may apply the Hopf lemma (Lemma A.3) at 0. From n w(0) = 0 and the Hopf lemma, it follows that w must be constant, hence identically zero. Remark 3.8. We present these two lemmas with U an open ball and an open half-ball respectively, since that is all we need to prove Aleksandrov s theorem. However, there is nothing special about these particular shapes; see Lemma A.3 and Lemma A.4 for weaker requirements on U. 4. Aleksandrov s method of moving planes In this section, we will prove the key step in Aleksandrov s theorem, which is the following statement. Proposition 4.1. Let S R n+1 be a connected, closed, CMC surface. Then S has a plane of symmetry in every direction. Let S R n+1 be a connected, closed, CMC surface. Let γ S n (where S n R n+1 is the unit sphere) and keep γ fixed throughout this section. Define P = {x R n+1 : x, γ = 0} (4.1) P t = {x R n+1 : x, γ = t} = P + tγ (for t R). (4.2) That is, P is the plane (i.e., n-dimensional subspace of R n+1 ) through the origin orthogonal to γ, and P t is the plane obtained by translating P a (signed) distance of t. For each t, P t divides S into two parts: S + t, the region above P t and S t, the region below P t. More precisely, S + t = S {x R n+1 : x, γ t} (4.3) S t = S {x R n+1 : x, γ t}. (4.4)

10 ALEKSANDROV S THEOREM: CLOSED SURFACES WITH CONSTANT MEAN CURVATURE 10 (See Figure 4.1.) Note that S + t = S t = S P t. S + t P t S t Figure 4.1. The plane P t divides S into S t + all following figures, γ is pointing up. and S t. In this figure and Since S is closed, it is the boundary of a bounded, closed region X R n+1. Let Ŝ t be the reflection of St through P t. For t very small (i.e., very negative), Ŝ t is empty since P t lies completely below S. As we gradually increase t, Ŝt becomes nonempty and eventually sticks out of X. Define t 0 = inf{t R : Ŝ t X}. (4.5) (See Figure 4.2 and Figure 4.3.) Note that t 0 is always finite since S is compact. S + t S + t S + t Ŝ t Ŝ t P t Ŝ t P t P t S t S t S t (a) t < t 0 (b) t = t 0 (c) t > t 0 Figure 4.2. Various values of t for a particular surface We set t to be t 0. For this particular value of t, at least one of the following must occur. Case 1: The intersection int(s + t ) int(ŝ t ) is nonempty. (Here, int stands for interior. ) This is the case in Figure 4.2b. Case 2: There is a point y S P t (recall S P t = S + t = Ŝ t ) such that the tangent planes at y to S + t and to Ŝ t coincide. This is the case in Figure 4.3b.

11 ALEKSANDROV S THEOREM: CLOSED SURFACES WITH CONSTANT MEAN CURVATURE 11 S + t S + t Ŝ t Ŝ t P t Ŝ t P t S + t P t S t S t S t (a) t < t 0 (b) t = t 0 (c) t > t 0 Figure 4.3. Various values of t for another surface In each of these two cases, we will show that P t is a plane of symmetry for S, i.e., S + t = Ŝ t First case: an interior point in common. As already established, t will be as in (4.5). By minimality of t we can write Ŝ t as a graph over the plane P t. We will use this fact in the following lemma. Lemma 4.2. If y int(s + t ) int(ŝ t ), then there is an open neighborhood of y in which the two surfaces are the same. Proof. Let y int(s + t ) int(ŝ t ). Choose a coordinate system (x 1,..., x n+1 ) with the following properties: The origin corresponds to the point y. The coordinates x 1,..., x n span the plane parallel to P. The coordinate x n+1 corresponds to the direction of γ. (See Figure 4.4.) In these coordinates, there is a function u in a neighborhood of 0 R n such that locally, Ŝ t is given by {x n+1 = u(x)}. (Note that u(0) = 0, since this corresponds to the point y.) Since y is in the interior of Ŝ t and Case 2 has not occurred for smaller values of t (by minimality of t), we know that u(0) exists. In other words, the normal to Ŝ t at y is not parallel to P t. Also by minimality of t, Ŝt lies below S t +, so S t + and Ŝ t have the same tangent plane at y. It follows that the normal to S t + at y is also not parallel to P t. Thus, we can find an open ball U R n centered at the origin such that S t + may also be written locally as a graph {(x, v(x))} for some v : U R (with v(0) = 0). u, v, D 2 u, D 2 v are bounded on U.

12 ALEKSANDROV S THEOREM: CLOSED SURFACES WITH CONSTANT MEAN CURVATURE 12 x 2 y x 1 P t Figure 4.4. Local coordinate system near y Since Ŝ t lies below S t +, u v on U. Also, u(y) = v(y). Thus, u = v on U, by Lemma 3.6. A consequence of the lemma is that int(s + t ) int(ŝ t ) is an open subset in int(ŝ t ). It is also closed subset 1 (since S + t Ŝ t is a closed subset of Ŝ t ). Thus, if y int(s + t ) int(ŝ t ), then the two surfaces agree on the connected component S of int(ŝ t ) that contains y. It follows that cl(s ) (the closure of S in R n+1 ) and its reflection are contained in S. Together, they form a connected closed surface and hence must be all of S. Thus, P t is a plane of symmetry Second case: a boundary point in common. Let y S P t be a point such that the tangent planes at y to S + t and to Ŝ t coincide. This implies the tangent plane to S at y contains the direction γ. Choose a coordinate system (x 1,..., x n+1 ) with the following properties: The origin corresponds to the point y. The coordinates x 1,..., x n span the plane tangent to S at y The coordinate x n corresponds to the direction of γ The coordinate x n+1 corresponds to the outward pointing normal to S at y. (See Figure 4.5.) With this choice of coordinates, S + t and Ŝ t can locally be described as graphs. More precisely, Ŝ t is given by x n+1 = u(x 1,..., x n ) and S + t is given by x n+1 = v(x 1,..., x n ) 1 We point out that there are two different meanings of closed used in these paragraphs. One is closed to describe a surface that is compact and without boundary. The other is in the topological sense.

13 ALEKSANDROV S THEOREM: CLOSED SURFACES WITH CONSTANT MEAN CURVATURE 13 x 1 x 2 y P t Figure 4.5. Local coordinate system near y for some u, v C 2 (U) C(Ū), where U = {x Rn : x 2 < r and x n > 0} for some r > 0. We can choose r small enough so that u, v, D 2 u, D 2 v are bounded on U. Since y Ŝ t S t +, we have u(0) = v(0) = 0. Since the normal vector to S at y lies in P t, we have n u(0) = n v(0) = 0. Minimality of t implies u v on U. Hence, by Lemma 3.7, u = v on U. Thus, int(s t + ) int(ŝ t ) is nonempty, so we can reduce ourselves to Case 1. This completes the proof of Proposition 4.1. Then using Lemma 1.4, we may deduce Aleksandrov s theorem: Corollary 4.3 (Aleksandrov s theorem). Let S R n+1 be a connected, closed, CMC surface. Then S is an n-sphere. 5. The Gaussian curvature case Recall that the principal curvatures of a surface S R n+1 are the eigenvalues of the second fundamental form II, and that the Gaussian curvature K of a surface S is the product of the principal curvatures. (Hence K = det II.) Aleskandrov s argument presented above can be modified to show that closed surfaces of constant Gaussian curvature must also be spheres. The rough outline is as follows. (1) A closed surface with constant Gaussian curvature must be strictly convex. (Lemma 5.5.) (2) If u : U R gives a surface of constant Gaussian curvature K, then u solves a particular nonlinear second-order PDE. (Corollary 5.2.) (3) The difference w of two such functions satisfies an SMP-admissible PDE, provided that the two surfaces are strictly convex. (Lemma 5.6.) (4) Because such w satisfy an SMP-admissible PDE, a closed surface with constant Gaussian curvature must have a plane of symmetry in each direction. (Aleksandrov s method of moving planes.)

14 ALEKSANDROV S THEOREM: CLOSED SURFACES WITH CONSTANT MEAN CURVATURE Gaussian curvature of graphs. Lemma 5.1. For a surface given by a graph {x n+1 = u(x)}, the Gaussian curvature is given (pointwise) by Proof. We compute (using (2.4) and (2.5)): K = (1 + u 2 ) (n+2)/2 det D 2 u. (5.1) II = (1 + u 2 ) 1/2 D 2 u (1 + u 2 ) 3/2 ( u)( u) T (D 2 u). (5.2) Here we view u as a column vector, so ( u)( u) T and D 2 u are both n n matrices. Factoring the expression and taking the determinant, the Gaussian curvature is ) K = det II = (1 + u 2 ) n/2 det (I ( u)( u)t det D 2 u. (5.3) 1 + u 2 To simplify this, we claim that for any column vector v R n, det(i vv T ) = 1 v 2. (5.4) It is clear that (5.1) follows from (5.3) and (5.4). The proof of (5.4) is by elementary linear algebra: without loss of generality, assume v 0. Because the matrix vv T has rank 1, it has 0 as an eigenvalue with multiplicity n 1. Since (vv T )v = v 2 v, v is an eigenvector with eigenvalue v 2. Thus, the eigenvalues of I vv T are 1 v 2 (multiplicity 1) and 1 (multiplicity n 1). Corollary 5.2. If the graph of u : U R has Gaussian curvature K : U R, then u satisfies F (D 2 u, u) = 0, where F (A, p) = det A K(1 + p 2 ) (n+2)/2. (5.5) Remark 5.3. The second-order PDE F (D 2 u, u) = 0 as above is nonlinear. We consider such PDE in Appendix B Surfaces of constant Gaussian curvature. Recall that we assume our surfaces are C 2. (See Remark 1.1.) Definition 5.4. A function u : U R is strictly convex if D 2 u is positive-definite on U. A surface is strictly convex if near every point, it can be written locally as a graph of a strictly convex function. We make the following preliminary observations about surfaces of constant Gaussian curvature (which do not depend on the results of Section 5.1). Lemma 5.5. If S is a closed surface of constant Gaussian curvature K, then K > 0 and S is strictly convex.

15 ALEKSANDROV S THEOREM: CLOSED SURFACES WITH CONSTANT MEAN CURVATURE 15 Proof. Since S is closed, there exists a closed ball B R n+1 of minimal volume that contains S. Let x S B. Because S B and S is tangent to B at x, the principal curvatures of S at x are strictly positive (since they cannot be less than the curvature of B). Since the Gaussian curvature is the product of the principal curvatures, it follows that the Gaussian curvature at x must be positive, so K > 0. Thus, the principal curvatures are positive everywhere (since K = 0 at a point where a principal curvature changes sign). Thus, S is strictly convex. Next, we combine Lemma 5.5 with the results of Section 5.1. First, we introduce some notation: for a matrix A and constants c < C, we write ci < A < CI to mean the eigenvalues of A are all strictly between c and C. Lemma 5.6. Suppose the graphs of u, v : U R both have constant Gaussian curvature K, and ci < D 2 u < CI and ci < D 2 v < CI (5.6) for some 0 < c < C. Then w := u v satisfies an SMP-admissible PDE. Proof. Both u and v satisfy the PDE F = 0, where F (A, p) = det A K(1 + p 2 ) (n+2)/2. (5.7) (This differs from (5.5) since here, K is a positive constant.) The matrix of derivatives ( F a ij ) i,j is given by ( ) F (A, p) = adj A, (5.8) a ij i,j where adj A is the adjugate matrix of A. Recall that A adj A = (det A)I, so there exist 0 < c < C such that ci < A < CI implies c I < adj A < C I. (5.9) Combining (5.6), (5.8), (5.9), and Lemma B.2, we see that w := u v satisfies an SMP-admissible PDE. Using Lemma 5.6, we obtain the analogues of the local uniqueness results in Section 3.3, so we can apply Aleksandrov s method of moving planes as in Section 4. This gives us the desired result: Theorem 5.7. Let S R n+1 be a connected, closed surface of constant Gaussian curvature. Then S is an n-sphere.

16 ALEKSANDROV S THEOREM: CLOSED SURFACES WITH CONSTANT MEAN CURVATURE 16 Appendix A. The Hopf lemma and the strong maximum principle Definition A.1. In this paper, a PDE with unknown w C 2 (U) is said to be SMPadmissible if the PDE is of the form a ij (x) ij w(x) + b k (x) k w(x) = 0 on U, (A.1) where k=1 the functions a ij, b k : U R are bounded. the matrix A(x) := (a ij (x)) is symmetric and uniformly elliptic. By uniformly elliptic, we mean that there is some constant λ > 0 such that for all x U, the smallest eigenvalue of A(x) is at least λ. Viewing A(x) as a symmetric bilinear form, this is also equivalent to (A(x))(ξ, ξ) λ ξ 2 for all x U and ξ R n. Remark A.2. SMP stands for strong maximum principle. Solutions to SMPadmissible PDE satisfy some hypotheses for the Hopf lemma (Lemma A.3) and the strong maximum principle (Lemma A.4). An SMP-admissible PDE is just a homogeneous linear second-order elliptic PDE with no zeroth order term, i.e., there is no c(x)w(x) term in Equation (A.1). The term SMP-admissible is non-standard; in fact, it is so non-standard that its only known use is in [Cha15]. Lemma A.3 (Hopf lemma). Let U be an open set. Suppose w C 2 (U) C(Ū) solves an SMP-admissible PDE. Suppose there exists a point y U such that w(y) > w(x) for all x U, and suppose that there is an open ball B U with y B. Then w ν (y) > 0, where ν is the outer unit normal to B at y. Lemma A.4 (Strong maximum principle). Let U be a connected, open, and bounded set. Suppose w C 2 (U) C(Ū) solves an SMP-admissible PDE. If w attains its maximum over Ū at an interior point, then w is constant within U. See [Eva10, Section 6.4] or [GT01, Chapter 3] for proofs and more details. Appendix B. Nonlinear second-order PDE We consider second-order PDE of the form F (D 2 u, u) = 0, for some smooth function F (A, p). Here, A = (a ij ) and p = (p 1,..., p n ), so F is a function R n n R n R. Note that such a PDE does not depend on the function u or on x U.

17 ALEKSANDROV S THEOREM: CLOSED SURFACES WITH CONSTANT MEAN CURVATURE 17 Definition B.1. Let U be a subset of R n n R n. We say F is uniformly elliptic on U if the matrix ( ) n F (B.1) a ij is uniformly elliptic on U. (See Definition A.1.) While nonlinear equations are in general harder than linear ones, the difference w := u v of two solutions u, v to a nonlinear PDE actually satisfies a linear PDE! The following lemma is a generalization of Lemma 3.2. Lemma B.2. Consider a second-order PDE given by F (A, p) = 0 and suppose u, v : U R are both solutions. Then w := u v satifies a linear second-order PDE. Furthermore, if F is uniformly ellptic on U := {(D 2 v + td 2 w, v + t w) : x U, t [0, 1]} R n n R n, (B.2) then the PDE satisfied by w is SMP-admissible. Proof. Write F [u] as short-hand for F (D 2 u, u). Suppose u, v : U R are both solutions to F, i.e., F [u] = F [v] = 0. Let w = u v. Then define G(t) for t R by G(t) = F [(1 t)v + tu] = F [v + tw]. (Note that for each t R, G(t) is a function U R.) We see that G(0) = G(1) = 0, so 1 0 G (t) dt = 0. By using the chain rule to expand G (t), 0 = i,j ( 1 0 F a ij [v + tw] dt ) ij w + ( 1 k=1 0 ) F [v + tw] dt k w p k (B.3) where F a ij [v + tw] denotes F a ij (D 2 v + td 2 w, v + t w) and likewise for F p k [v + tw]. This is a linear second-order PDE in w. Now, suppose furthermore that F is uniformly elliptic on U (defined by (B.2)). Then there is a λ > 0 such that the eigenvalues of ( ) n F [v + tw] (B.4) a ij are at least λ for all x U and t [0, 1]. It follows that the eigenvalues of ( 1 ) n F [v + tw] dt a ij 0 are at least λ for all x U; hence (B.3) is an SMP-admissible PDE. (B.5) Remark B.3. Just like in Lemma 3.2, the coefficients of the linear PDE (B.3) depend on u and v. (See Remark 3.3.)

18 ALEKSANDROV S THEOREM: CLOSED SURFACES WITH CONSTANT MEAN CURVATURE 18 Acknowledgments This exposition is my final project for Professor Amie Wilkinson s Riemannian geometry course (spring 2015). I would like to thank her for teaching the course and for giving us the opportunity to explore connections between Riemannian geometry and other fields. Super big thanks to Professor Luis Silvestre for advising me on this project throughout the quarter. He chose the topic, explained Aleksandrov s theorem to me, and put up with the millions of questions I had while writing this up. If anyone finds this paper helpful, it is thanks to him. 2 Thanks to Seung uk Jang and Minh-Tam Trinh for answering a question I had about Lemma 5.5. Thanks to Yu Li for pointing me to a reference for Remark 2.7. And thanks to all the other first-year graduate students for their support as we worked together through our first year in Chicago. References [Ale62] A. D. Aleksandrov, Uniqueness theorems for surfaces in the large. V, Amer. Math. Soc. Transl. (2) 21 (1962), MR (27 #698e) [BN91] H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. (N.S.) 22 (1991), no. 1, MR (93a:35048) [Cha15] A. Chang, Closed surfaces with constant mean curvature. [Eva10] Lawrence C. Evans, Partial differential equations, second ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, MR (2011c:35002) [GNN79] B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, MR (80h:35043) [GT01] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1998 edition. MR (2001k:35004) [Hop83] Heinz Hopf, Differential geometry in the large, Lecture Notes in Mathematics, vol. 1000, Springer-Verlag, Berlin, 1983, Notes taken by Peter Lax and John Gray, With a preface by S. S. Chern. MR (85b:53001) [Lee97] John M. Lee, Riemannian manifolds, Graduate Texts in Mathematics, vol. 176, Springer- Verlag, New York, 1997, An introduction to curvature. MR (98d:53001) [Ser71] James Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43 (1971), MR (48 #11545) University of Chicago 2 If anyone finds this paper unhelpful, I do not accept any responsibility.