Geometry of Lagrangian self-shrinking tori and applications to the Piecewise Lagrangian Mean Curvature Flow

TL;DR

This paper investigates the geometric properties of Lagrangian self-shrinking tori in four-dimensional space, establishing finiteness of entropy values, compactness results, and constructing a piecewise mean curvature flow that preserves Lagrangian conditions and avoids certain singularities.

Contribution

It introduces a gradient inequality for branched conformal self-shrinking tori, proves finiteness of entropy values, and constructs a piecewise Lagrangian mean curvature flow with singularity perturbation in 1 dimensions.

Findings

Finiteness of entropy values for bounded-area Lagrangian self-shrinking tori.

Compactness and embeddedness results for small-area tori.

Construction of a piecewise Lagrangian mean curvature flow avoiding certain singularities.

Abstract

We study geometric properties of the Lagrangian self-shrinking tori in $\mathbb R^4$. When the area is bounded above uniformly, we prove that the entropy for the Lagrangian self-shrinking tori can only take finitely many values; this is done by deriving a {\L}ojasiewicz-Simon type gradient inequality for the branched conformal self-shrinking tori and then combining with the compactness theorem in \cite{CMa}. When the area bound is small, we show that any Lagrangian self-shrinking torus in $\mathbb R^4$ with small area is embedded with uniform curvature estimates, and the space of such tori is compact. Using the finiteness of entropy values, we construct a piecewise Lagrangian mean curvature flow for Lagrangian immersed tori in $\mathbb R^4$, along which the Lagrangian condition is preserved, area is decreasing, and the type I singularities that are compact with a fixed area upper bound can be perturbed away in finite steps. This is a Lagrangian version of the construction for embedded surfaces in $\mathbb R^3$ in \cite{CM}.