Abstracts
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.
surfaces with boundary; constant mean curvature; elliptic partial differential equation
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.
superfícies com bordo; curvatura média constante; equações diferenciais parciais elípticas
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 nD 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 = nD. 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 Tp 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 C1, 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 W1, 2(W).
Let us denote the linear space of k-times weakly differentiable functions by Wk(W). For p ³ 1 and k a non-negative integer, we let
The Hölder spaces Ck,a(W) are defined as the subspaces of Ck(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 : = Di(aijDjv), v Î W1, 2(W) i, j = 1, 2
and
L2v : = AijDijv, v Î W2, 2(W) i, j = 1, 2,
where the coefficients aij are given by
and the coefficients Aij are defined by A11 = 1 + uy2, A12 = A21 = - uxuy, A22 = 1 + ux2. Finally, the symbols Di, Dij, 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 C1,a(W). Moreover, by the Lebesgue's dominated convergence theorem and Lemma 7.24 (Gilbarg and Trudinger 1983) we can conclude that u Î W2, p() for any subdomain ÌÌ W. Fixed ÌÌ W, we consider the equation
We observe that u Î W2, 2(). Thus, L2u is well-defined. Moreover, we have that in . It means that u Î W2, 2() is a solution to the equation (1) just above. Now using the Theorem 9.19 (Gilbarg and Trudinger 1983) we obtain u Î C2,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: hinojosa@mat.ufpb.br
- 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
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Publication in this collection
24 May 2002 -
Date of issue
Mar 2002
History
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Accepted
28 Nov 2001 -
Received
25 Oct 2001