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

Abstract

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 π, we show there exists a Lagrangian selfexpander 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 [1], Joyce [12], Lawlor [15] and Lee and Wang [18,19].

The second type singularities of symplectic and lagrangian mean curvature flows

“…The symplectic mean curvature flow was studied in [2-4, 10, 11, 22]. There are many references for Lagrangian mean curvature flows (see [8,[16][17][18][19][20][21]). …”

“…Symplectic or Lagrangian translating solitons were studied in [11,12,16,18] recently. In [11,12,18], some kinds of Liouville theorems were proved, and in [16], the authors constructed Lagrangian translating solitons.…”

Local non-collapsing of volume for the Lagrangian mean curvature flow

“…Note, that it was shown by Joyce, Lee and Tsui in [4] that for any ǫ ∈ (0, 1) there exist non trivial translating Lagrangian solitons which satisfy cos α ≥ ǫ.…”

Spacelike self-similar shrinking solutions of the mean curvature flow in pseudo-Euclidean spaces

“…Colding and Minicozzi [11] showed that the only stable selfshrinkers of the MCF for initial smooth closed embedded surfaces in R 3 are cylinders and spheres. For the Lagrangian MCF, Joyce et al [23], Anciaux [3] and Wang [29] have examples. There are other results in different contexts.…”