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

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

2008-01-24

arxiv.org/abs/0801.3721v2

Mathematics

Differential Geometry

33 pages. (v2) minor corrections. to appear in Journal of Differential Geometry

Scientific paper

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.

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