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

TL;DR

This paper investigates singularities in the Lagrangian mean curvature flow at critical phases, providing interior estimates, constructing viscosity solutions, and introducing new methods for regularity analysis.

Contribution

It establishes interior estimates for singularities at critical phases and develops novel techniques for $C^{2,eta}$ regularity in higher dimensions.

Findings

Interior estimates for singularities at critical phase

Construction of $C^{eta}$ viscosity solutions showing necessity of criticality

Introduction of a new method for $C^{2,eta}$ estimates via exponentiating the arctangent operator

Abstract

We establish interior estimates for singularities of the Lagrangian mean curvature flow when the Lagrangian phase is critical, i.e., $|\Theta|\geq (n-2)\tfrac{\pi}{2}$, and extend our results to the broader class of Lagrangian mean curvature type equations. Our gradient estimates require certain structural conditions, and we construct $C^{\alpha}$ 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 $C^{2,\alpha}$ estimates by exponentiating the arctangent operator into a concave one when $|\Theta|\geq (n-2)\tfrac{\pi}{2}$ and $n>2$.