Prescribed mean curvature hypersurfaces in \(H^{n+1}(-1)\) with convex planar boundary. I (Q1392731)

Immersed prescribed mean curvature compact hypersurfaces with boundary in \(H^{n+1}(-1)\) are studied in this work. Given a positive real number \(h_0\), a \(C^2\) hypersurface \(\Gamma\) of a hyperplane of the hyperbolic space is said to be \(h_0\)-convex if all its principal curvatures are greater than or equal to \(h_0\). One of the main results of this work is the following uniqueness theorem. Let \(P\) be a hyperplane of \(H^{n+1}(-1)\) and \(D\) a domain of \(P\) whose boundary is a smooth manifold \(\Gamma \) which is \(h_0\)-convex. Let \(M\) be a compact connected \(n\)-dimensional manifold with smooth boundary \(\partial M\) and \(x:M\to H^{n+1}(-1)\) an immersion with mean curvature \(H\) such that \(x| _{\partial M}\) is a diffeomorphism onto \(\Gamma\). Assume \(H\) is constant or is the restriction of a smooth function defined on a domain of \(H^{n+1}(-1)\) which depends only on the variables \(x_1,\ldots, x_n\) and that one of the following two conditions hold: (a) \(0<| H| \leq 1\) and \(h_0>1\), (b) \(h_0>1\), \(h_0>| H| >0\) and \(x(M)\) is contained in an open ball of the hyperbolic space whose boundary is a sphere of mean curvature \(h_0\). Then \(x(M)\) is the graph of a function \(g:\overline{D} \to R\).