Mean Curvature Flows of Lagrangian Submanifolds with Convex Potentials

Abstract

This article studies the mean curvature flow of Lagrangian submanifolds. In particular, we prove the following global existence and convergence theorem: if the potential function of a Lagrangian graph in T 2n is convex, then the flow exists for all time and converges smoothly to a flat Lagrangian submanifold. We also discuss various conditions on the potential function that guarantee global existence and convergence.

Mean curvature flows and isotopy of maps between spheres

“…We remark that a similar positive definite tensor has been considered for the Lagrangian mean curvature flow in Smoczyk [11] and Smoczyk and Wang [12]. The following lemma shows that the distance-decreasing condition is preserved by the mean curvature flow if k 1 ≥ |k 2 |.…”

“…Next we recall the evolution equation of parallel 2-tensors from [12]. The calculation indeed already appears in [14].…”

Mean Curvature Flow with Bounded Gauss Image

“…Recently, the mean curvature flow of higher codimension received much attention (see [2][3][4][17][18][19]25] and [26] for example). In this paper we investigate the mean curvature deformation of a complete space-like submanifold in pseudo-Euclidean space.…”

Perspectives on geometric analysis

“…For stable Lagrangian cycles, mean curvature flow should converge to special Lagrangian cycles. See M. T. Wang [701,702], Smoczyk [632] and Smoczyk-Wang [633]. The geometry of mirror symmetry was explained by Strominger-Yau-Zaslow in [639] using a family of special Lagrangian tori.…”