论文标题
双曲线空间中恒定曲率高度曲面的曲率估计值
Curvature estimates for hypersurfaces of constant curvature in hyperbolic space
论文作者
论文摘要
在本说明中，我们证明，对于每$ 0 <σ<1 $，存在一个平稳的完整hypersurface $σ$ in $ \ mathbb {h}^{n+1} $，带有规定的渐近边界$ \ partialσ=γ$ infinity in Infinity in Infinity in Infination in Infinity in Infinity in Infination in Infination in Infine满足$ f（κ）=σ$在$σ$的每个点处。以前，这个问题已由Guan-Spruck在[J.欧元。数学。 Soc。 （JEMS）12（2010），第1期。 3，797-817]，它们证明了$σ\ in（σ_0,1）$的存在结果，其中$σ_0> 0 $。我们证明的主要成分是对它们的精致曲率估计，当曲率函数$ f（κ）$具有可控的部分衍生物时，它适用，但这足以满足我们的目的；具体来说，我们解决了$ f = h_ {k}/h_ {k-1} $的问题，其中$ k $ - th的花园锥中$ h_k $是归一化的$ k $ - the $ k $ - the $ k $ - the $ k $ - the $ k $ - the thth ogq k \ leq k \ leq n $。
In this note, we prove that for every $0<σ<1$, there exists a smooth complete hypersurface $Σ$ in $\mathbb{H}^{n+1}$ with prescribed asymptotic boundary $\partial Σ=Γ$ at infinity, whose principal curvatures $κ=(κ_1,\ldots,κ_n)$ lie in a general cone $K$ and satisfy $f(κ)=σ$ at each point of $Σ$. Previously, the problem has been studied by Guan-Spruck in [J. Eur. Math. Soc. (JEMS) 12 (2010), no. 3, 797-817], and they proved the existence result for $σ\in (σ_0,1)$, where $σ_0>0$. A major ingredient of our proof is a refined curvature estimate of theirs that is applicable when the curvature function $f(κ)$ has controllable partial derivatives, but it is adequate for our purpose; specifically, we solve the problem for $f=H_{k}/H_{k-1}$ in the $k$-th Garding cone where $H_k$ is the normalized $k$-th elementary symmetric polynomial and $1 \leq k \leq n$.
