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arXiv 2302.01604math.AP
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双曲空间中具有预定曲率的凸超曲面

Convex hypersurfaces of prescribed curvatures in hyperbolic space

Li Chen

备注：
13 pages. arXiv admin note: substantial text overlap with arXiv:2301.01128
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摘要：

对于$\mathbb{H}^{n+1}$中光滑、闭、一致的$h$ -凸超曲面$M$，准球面高斯映射$G: M \rightarrow \mathbb{S}^n$是一个准同态.本文考虑了寻找一个光滑、闭、一致的$h$ -凸超曲面$M\subset \mathbb{H}^{n+1}$的问题，其$h$-次移动平均曲率$\widetilde{H}_{k}$（$1\leq k\leq n$）被定义为定义在$\mathbb{S}^n$上的一个正函数$\tilde{f}(x)$，即\begin{eqnarray*} \widetilde{H}_{k}(G^{-1}(x))=\tilde{f}(x). \end{eqnarray*}。如果给定的函数$\tilde{f}$是偶数，我们可以证明这个问题解的存在性。二十年前，Guan-Guan对欧氏空间中的凸超曲面也考虑了类似的问题。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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