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Some remarks on embedded hypersurfaces in hyperbolic space of constant curvature and spherical boundary. (English) Zbl 0831.53039
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.

MSC:
 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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References:
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