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Gauss curvature type flow and Alexandrov-Fenchel inequalities in the hyperbolic space

Tianci Luo, Rong Zhou

Abstract

We consider the Gauss curvature type flow for uniformly convex hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1}\ (n\geqslant 2)$. We prove that if the initial closed hypersurface is smooth and uniformly convex, then the smooth solution exists for all positive time and converges smoothly and exponentially to a geodesic sphere centered at the origin. The key step is to prove the upper bound of Gauss curvature and that uniform convexity preserves along the flow. As an application, we provide a new proof for an Alexandrov-Fenchel inequality comparing $(n-2)$th quermassintegral and volume of convex domains in $\mathbb{H}^{n+1}\ (n\geqslant 2)$.

Gauss curvature type flow and Alexandrov-Fenchel inequalities in the hyperbolic space

Abstract

We consider the Gauss curvature type flow for uniformly convex hypersurfaces in the hyperbolic space . We prove that if the initial closed hypersurface is smooth and uniformly convex, then the smooth solution exists for all positive time and converges smoothly and exponentially to a geodesic sphere centered at the origin. The key step is to prove the upper bound of Gauss curvature and that uniform convexity preserves along the flow. As an application, we provide a new proof for an Alexandrov-Fenchel inequality comparing th quermassintegral and volume of convex domains in .
Paper Structure (16 sections, 16 theorems, 130 equations)

This paper contains 16 sections, 16 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hypersurfaces in hyperbolic space
  4. Quermassintegrals
  5. Evolution equations
  6. Parametrization by radial graph
  7. $C^{0}$ and $C^{1}$ estimates
  8. $C^{0}$ Estimate
  9. $C^{1}$ Estimate
  10. $C^{2}$ estimate
  11. The upper bound of Gauss curvature $K$
  12. The bound of principal curvatures
  13. convergence
  14. Smooth Convergence
  15. Exponential convergence
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $M_{0}=X_{0}(\mathbb{S}^{n})$ be a smooth, closed, uniformly convex hypersurface in hyperbolic space $\mathbb{H}^{n+1}$ containing the origin, then the flow 1.1 has a unique smooth uniformly convex solution $M_{t}$ for all time $t\in[0,+\infty)$. When $t\rightarrow +\infty$, $M_{t}$ converges sm

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Corollary 2.2: Li2023AFA
  • proof
  • Lemma 2.3: Guan2013AMC
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • proof
  • ...and 18 more