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Zbl 0942.53044
Barbosa, João Lucas Marques; Sa Earp, Ricardo
Prescribed mean curvature hypersurfaces in $H^{n+1}$ with convex planar boundary. II. (English)
[A] Séminaire de théorie spectrale et géométrie. Année 1997-1998. St. Martin D'Hères: Université de Grenoble I, Institut Fourier, Sémin. Théor. Spectrale Géom., Chambéry-Grenoble. 16, 43-79 (1998).

The paper is the second part of the work presented in the Escola de Geometria held in Belo Horizonte, Brazil, in July 1998 [Geom. Dedicata 71, No. 1, 61-74 (1998; Zbl 0922.53023)]. \par Consider a hyperplane $P$ of $H^{n+1}$. Parametrize this space using the hyperplane model in such a way that $P$ is given by $x_0=0$. Given a domain $D$ in $P$ and a function $u:\overline D\mapsto\bbfR$, define the horizontal graph of $u$ in hyperbolic space by $G(u)=\{(u(x_1,\ldots ,x_n),x_1,\ldots ,x_n);(0,x_1, \ldots , x_n)\in \overline D\}$. A priori estimates for the solution to the Dirichlet problem for the mean curvature equation for horizontal graphs in hyperbolic space $H^{n+1}$ are established in the paper. These estimates are used to derive existence and uniqueness results. \par We formulate one of the results of the paper. Let $M$ be a compact connected $n$-dimensional manifold with smooth boundary and $x:M\to H^{n+1}(-1)$ an immersion with constant mean curvature $h$ whose boundary is a sphere $S^{n-1}(1)$ in a hyperplane of the hyperbolic space. \par (a) if $h=0$, then $M$ is the geodesic ball $D$ bounded by $S^{n-1}(1)$; \par (b) if $0<|h|<1$, then $M$ is a geodesic disc of an equidistant hypersurface; \par (c) if $|h|=1$, then $M$ is a geodesic disc of a horosphere; \par (d) if $|h|>1$ and $M$ is contained in ball of radius $r$ with $\coth r=|h|$, then $M$ is a geodesic disc of a sphere. \par For the existence theorems, the authors use some results of Schauder theory for elliptic quasilinear second order PDE's. Also, some modifications of the maximum principle are used to obtain a hyperbolic version of the graph lemma.
[Vladimir Tkachev (Volgograd)]

MSC 2000:: *53C42 Immersions (differential geometry)
53A10 Minimal surfaces, surfaces with prescribed mean curvature

Keywords: mean curvature equation; hypersurface; hyperbolic space; Hopf maximum principle; flux formula

Citations: Zbl 0922.53023

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