Surfaces of Constant Mean Curvature in Euclidean 3-space Orthogonal to a Plane along its Boundary

We consider compact surfaces with constant nonzero mean curvature whose boundary is a convex planar Jordan curve. We prove that if such a surface is orthogonal to the plane of the boundary, then it is a hemisphere.

Anais da Academia Brasileira de Ciências (2002) 74(1): 33-35 (Annals of the Brazilian Academy of Sciences) ISSN 0001-3765 www.scielo.br/aabc Let M be a compact surface inmersed in R 3 with constant mean curvature H whose boundary ∂M = is a planar Jordan curve of length L. Let D be a planar region enclosed by and let A be the area of D. Let us consider the cycle M ∪ D oriented in such a way that its orientation, along M, coincides with the one defined by the mean curvature vector. Let Y be a Killing vector field in R 3 and n D be a unitary vector field normal to D in the orientation of M ∪ D. Let ν be the unitary co-normal vector field along ∂M = pointing inwards M. By the flux formula it is known that |H | ≤ L 2A where equality holds if and only if ν = n D . That is, if and only if ν is constant and orthogonal to D along .
In this work we consider the case |H | ≤ L 2A and we show that, in the above conditions, if M is embedded and is convex, then M is a hemisphere. Explicitly we prove that: Theorem 1. Let M be a compact embedded surface in R 3 with constant mean curvature H = 0 whose boundary ∂M is a Jordan curve in a plane P ⊂ R 3 . Suposse that is convex and M is perpendicular to the plane P along its boundary. Then M is a hemisphere of radius 1 |H | .

PEDRO A. HINOJOSA
This theorem generalizes a result obtained by . We succed in discarding their assumption that ∂M should be a circle of radius 1 |H | . A sketch of the proof of the theorem is as follows. First, under the hypothesis of the theorem, M must be totally contained in one of the halfplanes determined by P (see , for example). Now let M * be the reflection of M with respect to the plane P. Since M is orthogonal to P along , we have that M := M ∪ M * is a compact surface without boundary, embedded in R 3 . Note that a priori M is only of class C 1 along . We will prove that M is at least of class C 3 . In this way we are able to use a classical result due to Alexandrov (see (Hopf 1983), for example) in order to establish that M is a sphere and therefore M is a hemisphere.
The regularity of M along is achieved by means of the theory of elliptic partial differential equations. Let p be any point in ⊂ M and be an open neighborhood of 0 in T p M chosen in such a way that locally around p, M may be described as the graph of a function u : → R. For our purposes, it is suffices to consider of class C 1,1 .
It is clear that u ∈ C 1 ( ). So, ∇u is well-defined and continuous. Since is bounded we have that u ∈ W 1,2 ( ).
Let us denote the linear space of k-times weakly differentiable functions by W k ( ). For p ≥ 1 and k a non-negative integer, we let W k,p ( ) = {u ∈ W k ( ), D σ u ∈ L p ( ) for all |σ | ≤ k}.
The Hölder spaces C k,α ( ) are defined as the subspaces of C k ( ) consisting of functions whose k-th order partial derivatives are locally Hölder continuous whith exponent α in .
We define on the following linear operators: where the coefficients a ij are given by a 11 = a 22 = 1 1 + |∇u| 2 , a 12 = a 21 = 0 and the coefficients A ij are defined by A 11 = 1 + u 2 y , A 12 = A 21 = −u x u y , A 22 = 1 + u 2 x . Finally, the symbols D i , D ij , i, j = 1, 2 stand for partial differentiation.
We prove that u is a weak solution to the equation L 1 u = 2H . By the Corollary 8.36 (Gilbarg and Trudinger 1983) we have u ∈ C 1,α ( ). Moreover, by the Lebesgue's dominated convergence theorem and Lemma 7.24 (Gilbarg and Trudinger 1983) we can conclude that u ∈ W 2,p ( ) for any subdomain ⊂⊂ . Fixed ⊂⊂ , we consider the equation (1)