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Geometric inequalities and their stabilities for modified quermassintegrals in hyperbolic space

Chaoqun Gao, Rong Zhou

Abstract

In this paper, we first consider the curve case of Hu-Li-Wei's flow for shifted principal curvatures of h-convex hypersurfaces in $\mathbb{H}^{n+1}$ proposed in [10]. We prove that if the initial closed curve is smooth and strictly h-convex, then the solution exists for all time and preserves strict h-convexity along the flow. Moreover, the evolving curve converges smoothly and exponentially to a geodesic circle centered at the origin. The key ingredient in our proof is the Heintze-Karcher type inequality for h-convex curves proved recently in [14]. As an application, we then provide a new proof of geometric inequalities involving weighted curvature integrals and modified quermassintegrals for h-convex curves in $\mathbb{H}^2$. We finally discuss the stability of these inequalities as well as Alexandrov-Fenchel type inequalities for modified quermassintegrals for strictly h-convex domains in $\mathbb{H}^{n+1}$.

Geometric inequalities and their stabilities for modified quermassintegrals in hyperbolic space

Abstract

In this paper, we first consider the curve case of Hu-Li-Wei's flow for shifted principal curvatures of h-convex hypersurfaces in proposed in [10]. We prove that if the initial closed curve is smooth and strictly h-convex, then the solution exists for all time and preserves strict h-convexity along the flow. Moreover, the evolving curve converges smoothly and exponentially to a geodesic circle centered at the origin. The key ingredient in our proof is the Heintze-Karcher type inequality for h-convex curves proved recently in [14]. As an application, we then provide a new proof of geometric inequalities involving weighted curvature integrals and modified quermassintegrals for h-convex curves in . We finally discuss the stability of these inequalities as well as Alexandrov-Fenchel type inequalities for modified quermassintegrals for strictly h-convex domains in .
Paper Structure (20 sections, 22 theorems, 179 equations)

This paper contains 20 sections, 22 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. Curvature flows
  3. Applications to geometric inequalities
  4. Stability results
  5. Preliminaries
  6. Elementary symmetric functions
  7. Quermassintegrals and modified quermassintegrals
  8. Hypersurfaces in hyperbolic space
  9. General evolution equations
  10. Flow for strictly h-convex curves in hyperbolic space
  11. A priori estimates
  12. Long time existence
  13. Smooth convergence
  14. Exponential convergence
  15. Geometric inequalities
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $X_0:M^n\to\mathbb{H}^{n+1}\ (n\geqslant 2)$ be a smooth embedding such that $M_0=X_0(M)$ is a smooth, strictly h-convex hypersurface in $\mathbb{H}^{n+1}$. Then the flow (equ-flowhy) has a smooth solution for all time $t\in[0,\infty)$, and $M_t=X_t(M)$ is strictly h-convex for each $t>0$ and co

Theorems & Definitions (34)

  • Theorem 1.1: Locallyconstrained
  • Theorem 1.2
  • Theorem 1.3: Li2022Hyperbolic
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6: DeRosa2018QuantitativeSF
  • Theorem 1.7
  • Lemma 2.1
  • Lemma 2.2: Newton-MacLaurin inequality
  • Lemma 2.3
  • ...and 24 more