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

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

PEDRO A. HINOJOSA

Centro de Ciências Exatas e da Natureza, Departamento de Matemática

Universidade Federal da Paraíba, JoãoPessoa, Brasil, and

Universidade Federal do Ceará, Centro de Ciências, Pós Graduação em Matemática

Campus do Pici, Bloco 914 - 60455-760 Fortaleza, Ce, Brasil

Manuscript received on October 25, 2001; accepted for publication on November 28, 2001;

presented by J. LUCAS BARBOSA

ABSTRACT

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.

Key words: surfaces with boundary, constant mean curvature, elliptic partial differential equation.

Let M be a compact surface inmersed in with constant mean curvature H whose boundary ¶M = G is a planar Jordan curve of length L. Let D be a planar region enclosed by G and let A be the area of D. Let us consider the cycle M È 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 and n_D be a unitary vector field normal to D in the orientation of M È D. Let Y be the unitary co-normal vector field along ¶M = G pointing inwards M. By the flux formula it is known that where equality holds if and only if v = n_D. That is, if and only if v is constant and orthogonal to D along G.

In this work we consider the case and we show that, in the above conditions, if M is embedded and G is convex, then M is a hemisphere. Explicitly we prove that:

THEOREM 1. Let M be a compact embedded surface in with constant mean curvature H ¹ 0 whose boundary ¶M is a Jordan curve G in a plane . Suposse that G is convex and M is perpendicular to the plane along its boundary. Then M is a hemisphere of radius

This theorem generalizes a result obtained by Brito and Earp (Brito and Earp 1991). We succed in discarding their assumption that ¶ M should be a circle of radius

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 (see (Brito et al. 1991), for example). Now let M^* be the reflection of M with respect to the plane . Since M is orthogonal to along G, we have that is a compact surface without boundary, embedded in . Note that a priori is only of class C¹ along G. We will prove that is at least of class C³. In this way we are able to use a classical result due to Alexandrov (see (Hopf 1983), for example) in order to establish that is a sphere and therefore M is a hemisphere.

The regularity of along G is achieved by means of the theory of elliptic partial differential equations. Let p be any point in G Ì and W be an open neighborhood of 0 in T_p chosen in such a way that locally around p, may be described as the graph of a function For our purposes, it is suffices to consider W of class C^{1, 1}.

It is clear that u Î C¹(W). So, u is well-defined and continuous. Since W is bounded we have that u Î W^{1, 2}(W).

Let us denote the linear space of k-times weakly differentiable functions by W^k(W). For p ³ 1 and k a non-negative integer, we let

The Hölder spaces C^k,a(W) are defined as the subspaces of C^k(W) consisting of functions whose k-th order partial derivatives are locally Hölder continuous whith exponent a in W.

We define on W the following linear operators:

L₁v : = D_i(a^ijD_jv), v Î W^{1, 2}(W) i, j = 1, 2

and

L₂v : = A^ijD_ijv, v Î W^{2, 2}(W) i, j = 1, 2,

where the coefficients a^ij are given by

and the coefficients A^ij are defined by A¹¹ = 1 + u_y², A¹² = A²¹ = - u_xu_y, A²² = 1 + u_x². 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 L₁u = 2H. By the Corollary 8.36 (Gilbarg and Trudinger 1983) we have u Î C^1,a(W). Moreover, by the Lebesgue's dominated convergence theorem and Lemma 7.24 (Gilbarg and Trudinger 1983) we can conclude that u Î W^{2, p}() for any subdomain ÌÌ W. Fixed ÌÌ W, we consider the equation

We observe that u Î W^{2, 2}(). Thus, L₂u is well-defined. Moreover, we have that in . It means that u Î W^{2, 2}() is a solution to the equation (1) just above. Now using the Theorem 9.19 (Gilbarg and Trudinger 1983) we obtain u Î C^2,a(). Repeating the same procedure we conclude that u Î (). Thus, is . So, is a regular compact closed surface embedded in with constant mean curvature. By the Theorem 5.2 (Chapter V, (Hopf 1983)) we conclude that is a (round) sphere and therefore M is a hemisphere.

ACKNOWLEDGMENTS

This work is part of my Doctoral Thesis at the Universidade Federal do Ceará - UFC. I want to thank my advisor, Professor J. Lucas Barbosa, and also Professor A. G. Colares, by the encouragement and many helpful conversations. The research was supported by a scholarship of CAPES.

RESUMO

Consideramos superfícies compactas com curvatura média constante e não nula as quais têm como bordo uma curva de Jordan plana convexa. Provamos que, se uma tal superfície é ortogonal ao plano do bordo então é um hemisfério.

Palavras-chave: superfícies com bordo, curvatura média constante, equações diferenciais parciais elípticas.

Correspondence to: Universidade Federal do Ceará

e-mail: [e-mail]

• BRITO F AND EARP RS. 1991. Geometric Configurations of Constant Mean Curvature Surfaces with Planar Boundary. An Acad bras Ci 63: 5-19.
• BRITO F, EARP RS, MEEKS W AND ROSENBERG H. 1991. Structure Theorems for Constant Mean Curvature Surfaces Bounded by a Planar Curve. Indiana Univ Math J 40(1): 333-343.
• GILBARG D AND TRUDINGER NS. 1983. Elliptic Partial Differential Equations of Second Order. 2nd edition, Springer-Verlag, Berlin.
• HOPF H. 1983. Differential Geometry in the Large. Lectures Notes in Mathematics, 1000, Springer-Verlag, Berlin.

Publication Dates

• Publication in this collection
  24 May 2002
• Date of issue
  Mar 2002

History

• Accepted
  28 Nov 2001
• Received
  25 Oct 2001