Convex hypersurfaces of prescribed curvatures in hyperbolic space
Li Chen

TL;DR
This paper proves the existence of smooth, uniformly h-convex hypersurfaces in hyperbolic space with prescribed k-th shifted mean curvature, extending Euclidean results to hyperbolic geometry under symmetry conditions.
Contribution
It establishes the existence of solutions for prescribed curvature problems in hyperbolic space, generalizing prior Euclidean results to a new geometric setting.
Findings
Existence of solutions when the prescribed function is even.
Extension of Euclidean convex hypersurface results to hyperbolic space.
Use of horospherical Gauss map in curvature prescription problems.
Abstract
For a smooth, closed and uniformly -convex hypersurface in , the horospherical Gauss map is a diffeomorphism. We consider the problem of finding a smooth, closed and uniformly -convex hypersurface whose -th shifted mean curvature () is prescribed as a positive function defined on , i.e. \begin{eqnarray*} \widetilde{H}_{k}(G^{-1}(x))=\tilde{f}(x). \end{eqnarray*} We can prove the existence of solution to this problem if the given function is even. The similar problem has been considered by Guan-Guan for convex hypersurfaces in Euclidean space two decades ago.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Analytic and geometric function theory · Point processes and geometric inequalities
