Longtime existence of the Lagrangian mean curvature flow

Abstract

Given a compact Lagrangian submanifold in flat space evolving by its mean curvature, we prove uniform C 2,α -bounds in space and C 2 -estimates in time for the underlying Monge-Ampère equation under weak and natural assumptions on the initial Lagrangian submanifold. This implies longtime existence and convergence of the Lagrangian mean curvature flow. In the 2-dimensional case we can relax our assumptions and obtain two independent proofs for the same result. (2000): 53C44 Mathematics Subject Classification

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 |.…”

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.…”

Uniqueness of unbounded solutions of the Lagrangian mean curvature flow equation for graphs

“…In fact, when u is a regular solution to (1), it is known that the graph (x, Du(x, t)) evolves by the mean curvature flow and it is a Lagrangian submanifold in R n × R n with the standard symplectic structure, for each t (cf. [5,6]). For a smooth stationary solution to (1), the graph of its gradient is a Lagrangian submanifold with zero mean curvature in R 2n .…”