Self-similar solutions and translating solitons for Lagrangian mean curvature flow

TL;DR

This paper constructs various self-similar and translating solitons for Lagrangian mean curvature flow, providing new models and insights into the flow's regularity and singularity formation.

Contribution

It introduces new classes of self-similar and translating solitons, including those with small oscillation in the Lagrangian angle, and establishes their role as local models for flow surgeries.

Findings

Existence of Lagrangian self-expanders asymptotic to pairs of transverse planes.

Construction of translating solitons analogous to cigar solitons in Ricci flow.

Families of self-shrinkers and self-expanders with different topologies.

Abstract

We construct many self-similar and translating solitons for Lagrangian mean curvature flow, including self-expanders and translating solitons with arbitrarily small oscillation on the Lagrangian angle. Our translating solitons play the same role as cigar solitons in Ricci flow, and are important in studying the regularity of Lagrangian mean curvature flow. Given two transverse Lagrangian planes R^n in C^n with sum of characteristic angles less than pi, we show there exists a Lagrangian self-expander asymptotic to this pair of planes. The Maslov class of these self-expanders is zero. Thus they can serve as local models for surgeries on Lagrangian mean curvature flow. Families of self-shrinkers and self-expanders with different topologies are also constructed. This paper generalizes the work of Anciaux, Joyce, Lawlor, and Lee and Wang.