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A Priori Estimates for Singularities of the Lagrangian Mean Curvature Flow with Supercritical Phase

Arunima Bhattacharya, Jeremy Wall

Abstract

In this paper, we prove interior a priori estimates for singularities of the Lagrangian mean curvature flow assuming the Lagrangian phase is supercritical. We prove a Jacobi inequality that holds good when the Lagrangian phase is critical and supercritical. We further extend our results to a broader class of Lagrangian mean curvature type equations.

A Priori Estimates for Singularities of the Lagrangian Mean Curvature Flow with Supercritical Phase

Abstract

In this paper, we prove interior a priori estimates for singularities of the Lagrangian mean curvature flow assuming the Lagrangian phase is supercritical. We prove a Jacobi inequality that holds good when the Lagrangian phase is critical and supercritical. We further extend our results to a broader class of Lagrangian mean curvature type equations.
Paper Structure (6 sections, 11 theorems, 63 equations)

This paper contains 6 sections, 11 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Jacobi inequality
  4. Lewy-Yuan rotation
  5. Gradient estimate
  6. Proof of the main results

Key Result

Theorem 1.1

If $u$ is a smooth solution of any of these equations: s, tran, and rotator on $B_{R}(0)\subset \mathbb{R}^{n}$ where $n\geq 2$ and $|\Theta|\geq (n-2)\frac{\pi}{2}+ \delta$, then we have where $C_1$ and $C_2$ are positive constants depending on $n$ and the following: $s_2$ for s, $t_2,t_3$ for tran, and $r_2$ for rotator.

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Lemma 2.1
  • proof
  • Proposition 2.1
  • Lemma 3.1
  • proof
  • Corollary 3.1
  • proof
  • ...and 11 more