Type II Singularities of Lagrangian Mean Curvature Flow with Zero Maslov Class

TL;DR

The paper addresses Type II singularities in Lagrangian mean curvature flow with zero Maslov class or almost calibrated conditions by analyzing blow-up limits and translating solitons in arbitrary dimension. It develops a novel cutoff-based, angle-aware method to derive rigidity estimates from the evolution equations $H=J\nabla\theta$ and $(\partial_t-\Delta)\theta=0$, yielding bounds that relate the mean curvature to the oscillation of the Lagrangian angle. The main contributions include a sharp bound $h^2\le\eta^2$ on $|H|^2$ in terms of the Lagrangian angle oscillation, a nonexistence result for certain eternal translators under $\cos\theta\ge\delta$ and $|H|^2\ge\varepsilon|A|^2$ with $\varepsilon>1-\delta$, and rigidity results for translating solitons with zero Maslov class under Euclidean or weighted polynomial area growth, including minimality and, in low dimension, flatness. These results extend known two-dimensional rigidity phenomena to arbitrary dimension and provide structural constraints on potential blow-up models for Type II singularities in Lagrangian flows.

Abstract

In this paper, we will prove some rigidity theorems for blow up limits to Type II singularities of Lagrangian mean curvature flow with zero Maslov class or almost calibrated Lagrangian mean curvature flows, especially for Lagrangian translating solitons in any dimension. These theorems generalized previous corresponding results from two dimensional case to arbitrarily dimensional case.

Theorems & Definitions (10)

• Theorem 1
• Theorem 2
• Corollary 3
• Theorem 4
• Theorem 5
• Remark 6
• Corollary 7
• Definition 8
• Theorem 9
• Corollary 10