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Curvature estimates for the level sets of solutions of the Monge-Ampère equation $\det D^2 u=1$

Chuanqiang Chen, Xi-Nan Ma, Shujun Shi

Abstract

For the Monge-Ampère equation $\det D^2 u=1$, we find new auxiliary curvature functions which attain respective maximum on the boundary. Moreover, we obtain the upper bounded estimates for the Gauss curvature and mean curvature of the level sets for the solution to this equation.

Curvature estimates for the level sets of solutions of the Monge-Ampère equation $\det D^2 u=1$

Abstract

For the Monge-Ampère equation , we find new auxiliary curvature functions which attain respective maximum on the boundary. Moreover, we obtain the upper bounded estimates for the Gauss curvature and mean curvature of the level sets for the solution to this equation.

Paper Structure

This paper contains 4 sections, 4 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['t1.1']}
  3. Proof of Theorem \ref{['t1.2']}
  4. Proof of the Corollary \ref{['c1']}

Key Result

Theorem \oldthetheorem

Let $\Omega \subset \mathbb R^n$ be a bounded convex domain, $n\geq 2$, $u$ the strictly convex solution of 61.1. Then the function attains its maximum on the boundary $\partial \Omega$.

Theorems & Definitions (11)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • proof
  • Remark \oldthetheorem
  • ...and 1 more