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Hessian estimates for shrinkers, expanders, translators, and rotators of the Lagrangian Mean Curvature Flow

Arunima Bhattacharya, Jeremy Wall

Abstract

In this paper, we prove interior Hessian estimates for shrinkers, expanders, translators, and rotators of the Lagrangian mean curvature flow under the assumption that the Lagrangian phase is hypercritical. We further extend our results to a broader class of Lagrangian mean curvature type equations.

Hessian estimates for shrinkers, expanders, translators, and rotators of the Lagrangian Mean Curvature Flow

Abstract

In this paper, we prove interior Hessian estimates for shrinkers, expanders, translators, and rotators of the Lagrangian mean curvature flow under the assumption that the Lagrangian phase is hypercritical. We further extend our results to a broader class of Lagrangian mean curvature type equations.
Paper Structure (6 sections, 12 theorems, 94 equations)

This paper contains 6 sections, 12 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The slope as a subsolution to a fully nonlinear PDE
  4. Sobolev Inequalities and the Lewy-Yuan rotation
  5. Proof of the main Theorems
  6. Appendix

Key Result

Theorem 1.1

If $u$ is a $C^4$ solution of any of these equations: s, tran, and rotator on $B_{R}(0)\subset \mathbb{R}^{n}$ where $|\Theta|\geq (n-1)\frac{\pi}{2}$, then we have where $C_1$ and $C_2$ are positive constants depending on $n$ and the following:

Theorems & Definitions (26)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Proposition 3.1
  • proof
  • Lemma 3.1
  • proof
  • Corollary 3.1
  • proof
  • ...and 16 more