Convex hypersurfaces of prescribed curvatures in hyperbolic space

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 $h$-convex hypersurface $M$ in $\mathbb{H}^{n+1}$, the horospherical Gauss map $G: M \rightarrow \mathbb{S}^n$ is a diffeomorphism. We consider the problem of finding a smooth, closed and uniformly $h$-convex hypersurface $M\subset \mathbb{H}^{n+1}$ whose $k$-th shifted mean curvature $\widetilde{H}_{k}$ ($1\leq k\leq n$) is prescribed as a positive function $\tilde{f}(x)$ defined on $\mathbb{S}^n$, 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 $\tilde{f}$ is even. The similar problem has been considered by Guan-Guan for convex hypersurfaces in Euclidean space two decades ago.