Translating Solitons to a Lagrangian mean curvature flow with zero Maslov class

TL;DR

This work analyzes Type II singularities in the Lagrangian mean curvature flow with zero Maslov class on Calabi–Yau manifolds. It introduces a new weighted monotonicity formula tied to the Lagrangian angle, and derives a necessary condition that any eternal blow-up limit must satisfy. The authors apply these tools to the Joyce–Lee–Tsui translating solitons, showing that under a finite-energy-type assumption on the angle near the singularity, these solitons cannot occur as blow-up limits for $n\ge2$. This narrows the space of possible singularity models for zero-Maslov LMCF and informs the open questions of Joyce–Lee–Tsui and Neves–Tian. The results provide a framework to test translating solitons as singularity models via monotonicity-based energy inequalities and blow-up analysis.

Abstract

It is known that there is no a Type I singularity for the Lagrangian mean curvature flow with zero Maslov class. In this paper, we study translating solitons which are important models of Type II singularities. A necessary condition for a blow-up limit arising at a Type II singularity of a Lagrangian mean curvature flow with zero Maslov class is provided. As an application, we try to understand the important open question proposed by Joyce-Lee-Tsui and Neves-Tian, whether the Lagrangian translating solitons constructed by Joyce-Lee-Tsui can be a blow-up limit for a Lagrangian mean curvature flow with zero Maslov class.

Theorems & Definitions (19)

• Definition 1.1
• Theorem 1.1
• Remark 1.2
• Proposition 2.1
• Proposition 3.1
• proof
• Corollary 3.2
• Theorem 4.1
• proof
• Corollary 4.2