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Curvature estimates for hypersurfaces of constant curvature in hyperbolic space II

Bin Wang

TL;DR

The work addresses the existence of smooth complete hypersurfaces in $\mathbb{H}^{n+1}$ with constant curvature given by symmetric functions of the hyperbolic principal curvatures and prescribed asymptotic boundary data. It advances the curvature-estimate backbone by deriving global $C^2$ bounds for two important classes: (i) the $(n-2)$-curvature equation with $f=H_{n-2}^{1/(n-2)}$ and (ii) quotient-type equations $f=(H_k/H_l)^{1/(k-l)}$, under suitable convexity and boundary hypotheses. The authors obtain a sharp curvature bound of the form $\max_{\Sigma} \kappa_{\max} \le C(1+\max_{\partial\Sigma}\kappa_{\max})$ by deploying the Guan–Spruck framework, refined concavity inequalities (Ren–Wang), and careful analysis of the associated fully nonlinear PDE, including a maximum-principle argument on a suitably chosen test function. These curvature estimates pave the way for existence results for all $\sigma\in(0,1)$ in the considered cones $K_k$, significantly extending prior results that required restricted ranges of $\sigma$, and they articulate a concrete path toward a general quotient-cone existence theory in hyperbolic space.

Abstract

In this note, we investigate the existence of smooth complete hypersurfaces in hyperbolic space with constant curvature and a prescribed asymptotic boundary at infinity. By deriving curvature estimates, we are able to deduce the existence in some cases. Previously, these existence results were proved for a restricted range of curvature values, while here we prove the existence for all possible curvature values.

Curvature estimates for hypersurfaces of constant curvature in hyperbolic space II

TL;DR

The work addresses the existence of smooth complete hypersurfaces in with constant curvature given by symmetric functions of the hyperbolic principal curvatures and prescribed asymptotic boundary data. It advances the curvature-estimate backbone by deriving global bounds for two important classes: (i) the -curvature equation with and (ii) quotient-type equations , under suitable convexity and boundary hypotheses. The authors obtain a sharp curvature bound of the form by deploying the Guan–Spruck framework, refined concavity inequalities (Ren–Wang), and careful analysis of the associated fully nonlinear PDE, including a maximum-principle argument on a suitably chosen test function. These curvature estimates pave the way for existence results for all in the considered cones , significantly extending prior results that required restricted ranges of , and they articulate a concrete path toward a general quotient-cone existence theory in hyperbolic space.

Abstract

In this note, we investigate the existence of smooth complete hypersurfaces in hyperbolic space with constant curvature and a prescribed asymptotic boundary at infinity. By deriving curvature estimates, we are able to deduce the existence in some cases. Previously, these existence results were proved for a restricted range of curvature values, while here we prove the existence for all possible curvature values.
Paper Structure (4 sections, 9 theorems, 90 equations)

This paper contains 4 sections, 9 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The $(n-2)$-curvature equation
  4. Quotient equations with curvature bounded from below

Key Result

Theorem 1.3

Let $n \geq 5$. Given a disjoint collection of closed embedded smooth $(n-1)$-dimensional submanifolds $\Gamma=\{\Gamma_1,\ldots,\Gamma_m\} \subseteq \partial_{\infty} \mathbb{H}^{n+1}$ and a constant $0<\sigma<1$, if $\Gamma=\partial \Omega$ is mean-convex, then there exists a smooth complete stric with the asymptotic boundary at infinity.

Theorems & Definitions (19)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Conjecture 1.8
  • Lemma 2.1
  • Lemma 2.2
  • ...and 9 more