Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Singularities of the Lagrangian mean curvature flow at the critical Lagrangian phase

Arunima Bhattacharya, Ravi Shankar, Jeremy Wall, Diego Yepez

TL;DR

The paper addresses regularity for the Lagrangian mean curvature flow under a critical Lagrangian phase, establishing gradient and Hessian estimates when $|\Theta| \ge (n-2)\frac{\pi}{2}$. It develops a gradient-estimation framework under structural conditions on $\Theta$, proves Hessian bounds via a Jacobi-type analysis and MVI arguments, and introduces a new $C^{2,\alpha}$ approach by exponentiating the arctangent to a concave operator for $n>2$. It also constructs singular $C^{\alpha}$ viscosity solutions to demonstrate the necessity of the phase-criticality and the gradient-structure assumptions, and discusses limitations in low dimensions along with an alternate rotation-based route for $n=2$. Collectively, these results advance the regularity theory for LMCF and related Lagrangian-type PDEs, clarifying when interior smoothness can be guaranteed near singularities. The methods provide new analytical tools—especially the exponentiation technique for concavity—and broaden the applicability to a broader class of Lagrangian mean curvature type equations.

Abstract

We establish interior estimates for singularities of the Lagrangian mean curvature flow when the Lagrangian phase is critical, i.e., $|Θ|\geq (n-2)\tfracπ{2}$, and extend our results to the broader class of Lagrangian mean curvature type equations. Our gradient estimates require certain structural conditions, and we construct $C^α$ singular viscosity solutions to show that criticality of the phase is necessary, and that these conditions cannot be removed in dimension one. We also introduce a new method for proving $C^{2,α}$ estimates by exponentiating the arctangent operator into a concave one when $|Θ|\geq (n-2)\tfracπ{2}$ and $n>2$.

Singularities of the Lagrangian mean curvature flow at the critical Lagrangian phase

TL;DR

The paper addresses regularity for the Lagrangian mean curvature flow under a critical Lagrangian phase, establishing gradient and Hessian estimates when . It develops a gradient-estimation framework under structural conditions on , proves Hessian bounds via a Jacobi-type analysis and MVI arguments, and introduces a new approach by exponentiating the arctangent to a concave operator for . It also constructs singular viscosity solutions to demonstrate the necessity of the phase-criticality and the gradient-structure assumptions, and discusses limitations in low dimensions along with an alternate rotation-based route for . Collectively, these results advance the regularity theory for LMCF and related Lagrangian-type PDEs, clarifying when interior smoothness can be guaranteed near singularities. The methods provide new analytical tools—especially the exponentiation technique for concavity—and broaden the applicability to a broader class of Lagrangian mean curvature type equations.

Abstract

We establish interior estimates for singularities of the Lagrangian mean curvature flow when the Lagrangian phase is critical, i.e., , and extend our results to the broader class of Lagrangian mean curvature type equations. Our gradient estimates require certain structural conditions, and we construct singular viscosity solutions to show that criticality of the phase is necessary, and that these conditions cannot be removed in dimension one. We also introduce a new method for proving estimates by exponentiating the arctangent operator into a concave one when and .
Paper Structure (13 sections, 11 theorems, 89 equations)

This paper contains 13 sections, 11 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Jacobi Inequality
  4. Proof of the Gradient Estimates
  5. Singular Non-Lipschitz Viscosity Solutions
  6. Fails condition (0.3)
  7. Fails condition (0.2)
  8. Fails conditions (0.4) and (0.2)
  9. Higher dimensions
  10. Remarks
  11. Proof of the Hessian Estimates
  12. $C^{2,\alpha}$ Regularity
  13. Discussion: alternate approaches

Key Result

Theorem 1.1

If $u$ is a smooth solution of slag on $B_R\subset\mathbb{R}^n$, with $n\geq 3$ and $|\Theta|\geq (n-2)\frac{\pi}{2}$ where $\Theta(x,z,p)\in C^2(\Gamma_R)$ satisfies struct and is partially convex in $p$, then we get where $C_1$ and $C_2$ are positive constants depending on $\nu_1,\nu_2$, and $n$.

Theorems & Definitions (22)

  • Theorem 1.1: Hessian Estimates
  • Remark 1.1
  • Corollary 1.1
  • Remark 1.2
  • Theorem 1.2: Gradient Estimates
  • Remark 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Corollary 2.1
  • Proposition 2.1
  • ...and 12 more