Existence and topological uniqueness of compact CMC hypersurfaces with boundary in hyperbolic space (Q375745)

Let \(\mathbb H^{n+1}\) denote the hyperbolic space of dimension \(n+1\), which is the unique complete manifold of constant sectional curvature \(-1\). Let \(\mathbb H^n\) be a totally geodesic subspace of \(\mathbb H^{n+1}\) of dimension \(n\). The authors show that if \(\Gamma\) is a compact embedded hypersurface of dimension \(n-1\) of \(\mathbb H^n\) which satisfies the enclosing \(H\)-hypersphere condition with \(|H|<1\), then there is a unique compact (up to reflection) embedded constant mean curvature \(H\)-hypersurface \(M\) with boundary \(\Gamma\); in this setting \(M\) is diffeomorphic to a ball.