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An isoperimetric type inequality in De Sitter space

Ling Xiao

TL;DR

The paper addresses a sharp isoperimetric-type inequality for spacelike, star-shaped, and $2$-convex hypersurfaces in de Sitter space by designing and analyzing a curvature flow of the form $X_t=(u-b_{n,2}\phi'\sigma_2^{-1/2})\nu$. It establishes monotonicity of quermassintegrals, proves long-time existence with $C^{\infty}$ bounds, and demonstrates convergence to a radial coordinate slice, from which the main inequality follows. The key result is the inequality $\int_M \sigma_2 d\mu_g-(n-1)|M| \le \xi_{2,0}(|M|)$, with equality iff $M$ is a radial coordinate slice. The work extends classical Minkowski-type and Alexandrov-Fenchel-type inequalities to the de Sitter setting under 2-convexity, using a curvature-flow framework informed by Newton-Maclaurin inequalities and Hessian concavity to obtain sharp geometric control in a Lorentzian ambient space.

Abstract

In this paper, we prove an optimal isoperimetric inequality for spacelike, compact, star-shaped, and $2$-convex hypersurfaces in de Sitter space.

An isoperimetric type inequality in De Sitter space

TL;DR

The paper addresses a sharp isoperimetric-type inequality for spacelike, star-shaped, and -convex hypersurfaces in de Sitter space by designing and analyzing a curvature flow of the form . It establishes monotonicity of quermassintegrals, proves long-time existence with bounds, and demonstrates convergence to a radial coordinate slice, from which the main inequality follows. The key result is the inequality , with equality iff is a radial coordinate slice. The work extends classical Minkowski-type and Alexandrov-Fenchel-type inequalities to the de Sitter setting under 2-convexity, using a curvature-flow framework informed by Newton-Maclaurin inequalities and Hessian concavity to obtain sharp geometric control in a Lorentzian ambient space.

Abstract

In this paper, we prove an optimal isoperimetric inequality for spacelike, compact, star-shaped, and -convex hypersurfaces in de Sitter space.

Paper Structure

This paper contains 14 sections, 16 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Background and motivations
  3. Outline
  4. Preliminary
  5. Elementary symmetric functions
  6. Star-shaped graph in $\mathbb S^{n+1}_1$
  7. Hypersurfaces in $\mathbb S^{n+1}_1$
  8. Curvature flow and monotonicity formula
  9. A specific flow equation.
  10. Long time existence of \ref{['1.1']}
  11. Estimates up to first order
  12. Uniform bounds of $F$
  13. Uniform bounds for principal curvatures
  14. Convergence and inequality

Key Result

Theorem 1.2

Let $M_0\subset\mathbb S^{n+1}_1$ be a spacelike, compact, star-shaped, and strictly $2$-convex hypersurface. Then the solution to the following flow equation exists for all time, where $b_{n, 2}=(\sigma_2(I))^{1/2}=\left[\frac{n(n-1)}{2}\right]^{1/2}.$ Moreover, the flow hypersurfaces $M_t$ converge smoothly to a radial coordinate slice as $t\rightarrow\infty.$

Theorems & Definitions (26)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 3.1
  • ...and 16 more