Some remarks on embedded hypersurfaces in hyperbolic space of constant curvature and spherical boundary (Q1891974)

Let \(M\) be a compact embedded hypersurface with boundary \(C = \partial M\) in the hyperbolic space \(H^{m + 1}\) of constant curvature \(-1\) and denote the \(r\)th mean curvature of \(M\) by \(H_r\). The authors prove: a) If \(C\) is a sphere and \(H_r\) is a constant \(\in [0,1]\) for some \(r\), then \(M\) is part of a equidistant sphere of \(H^{m+1}\). b) If \(H_1\) is constant, \(C\) is a convex submanifold of a geodesic hyperplane \(N \subset H^{m + 1}\), and if \(M\) is transverse to \(N\) along \(C\), then \(M\) has all the symmetries of \(C\). For the last result a flux formula for Killing vector fields of the hyperbolic space is derived.