Translating Solitons to Symplectic and Lagrangian Mean Curvature Flows

Abstract

Abstract. In this paper, we construct finite blow-up examples for symplectic mean curvature flows and we study properties of symplectic translating solitons. We prove that, the Kähler angle α of a symplectic translating soliton with max |A| = 1 satisfies that sup |α| > π 4 |T | |T |+1 where T is the direction in which the surface translates.

Translating Solitons for Lagrangian Mean Curvature Flow in Complex Euclidean Plane

“…To prove the last part of the lemma, we define f := α ′ , α and g := α ′ , Jα and so κ α = bf − ag. Using that f ′ = 1 − κ α g, g ′ = κ α f and f 2 + g 2 = |α| 2 , it is only a long computation to check that κ α satisfies (11).…”

“…We know from Proposition 2 that its induced metric is conformal and, using the expression of its Lagrangian angle map, we get that α ⋆ ω is Hamiltonian stationary if and only if κ ′ α +κ ω = 0. Using (11) we obtain that κ α ≡ c 1 ∈ R and κ ω ≡ c 2 ∈ R such that (37) c 2 1 (c 2 1 − cos ϕ) = 0 = c 2 2 (c 2 2 + cos ϕ). If c 1 = 0, α must be a line and this implies that ϕ = 0 and, following the notation of Corollary 1, α = α 0 .…”

Symplectic mean curvature flows in Kähler surfaces with positive holomorphic sectional curvatures

“…In this paper we mainly concentrated on the symplectic mean curvature flows, which were studied by Chen -Tian [5], Chen-Li [3], Chen-Li-Tian [4], Wang [23], Han-Li [9], [10], [11], Han-Sun [18], Han-Li-Sun [13], and Han-Li-Yang [14]. The basic fact is that the symplectic property is preserved by the mean curvature flow if the ambient space M is Kähler-Einstein, or if the ambient Kähler surface evolves along the Kähler-Ricci flow [11].…”

The second type singularities of symplectic and lagrangian mean curvature flows

“…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.…”