Convex hypersurfaces of prescribed curvatures in hyperbolic space

Abstract

For a smooth, closed and uniformly $h$-convex hypersurface $M$ in ${\mathbb{H}}^{n+1}$, the horospherical Gauss map $G:M\to {\mathbb{S}}^n$ 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 ${\tilde{H}}_k$ ($1\le k\le n$) 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.

arXiv comment13 pages. arXiv admin note: substantial text overlap with arXiv:2301.01128