Translating solitons to symplectic mean curvature flows

Abstract

In this article, we prove some nonexistence results for the translating solitons to the symplectic mean curvature flows or to the almost calibrated Lagrangian mean curvature flows under some curvature assumptions.

The second type singularities of symplectic and lagrangian mean curvature flows

“…On the other hand, applying the techniques used in [12], we can rule out the existence of type II blow-up flows for a symplectic mean curvature flow which are normal flat. More precisely, we prove the theorem below.…”

“…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.…”

Non-existence of eternal solutions to Lagrangian mean curvature flow with non-negative Ricci curvature

“…Our aim in this paper is to show a non-existence result for noncompact complete eternal solutions of almost calibrated Lagrangian mean curvature flow with an additional curvature condition. In this direction, we know some non-existence results by Han-Sun [HS10], Neves-Tian [NT13] and Sun [Sun13] for translating solutions. In [HS10], Han-Sun showed a non-existence result for almost calibrated Lagrangian translating solitons with non-negative sectional curvatures.…”

“…In this direction, we know some non-existence results by Han-Sun [HS10], Neves-Tian [NT13] and Sun [Sun13] for translating solutions. In [HS10], Han-Sun showed a non-existence result for almost calibrated Lagrangian translating solitons with non-negative sectional curvatures. We generalize their theorem [HS10] to the class of almost calibrated Lagrangian eternal solutions with non-negative Ricci curvatures.…”

“…Our result is a parabolic version of the theorem by Han-Sun [HS10]. In order to show the parabolic version of the curvature estimate, we adopt techniques developed by Souplet-Zhang [SZ06] and Wang [Wan11].…”