Lagrangian mean curvature flow of pinched submanifolds of CP^n

TL;DR

This paper studies the evolution of Lagrangian submanifolds in complex projective space under mean curvature flow, showing that certain pinched initial conditions lead to convergence to totally geodesic submanifolds and topological classification.

Contribution

It establishes long-time existence and convergence results for Lagrangian mean curvature flow under pinching conditions in CP^n, providing new insights into their geometric and topological structure.

Findings

Flow exists for all time under pinching conditions

Submanifolds converge to totally geodesic submanifolds

Lagrangian submanifolds are diffeomorphic to real projective space

Abstract

We consider the evolution by mean curvature flow of Lagrangian submanifolds of the complex projective space CP^n. We prove that, if the initial value satisfies a suitable pinching condition, then the flow exists for all times and the manifold converges to a totally geodesic submanifold. As a corollary, we obtain that a Lagrangian submanifold satisfying our pinching condition is diffeomorphic to a real projective space.