A Note on the Evolution of the Whitney Sphere Along Mean Curvature Flow

Celso Viana

doi:10.1007/s12220-020-00432-z

Abstract

We study the evolution of the Whitney sphere along the Lagrangian mean curvature flow. We show that equivariant Lagrangian spheres in {{mathbb {C}}^n} satisfying mild geometric assumptions collapse to a point in finite time and the tangent flows converge to a Lagrangian plane with multiplicity two.

Highlights

The Whitney sphere is the immersion F : Sn → R2n given byF(x1, . . . , xn+1) = + 1 xn2+1 (x1 x1 xn+1, xn xn+1).This immersion is Lagrangian, i.e., F∗ω = 0, where ω is the standard symplectic form on R2n
From the point of view of topology, the Whitney sphere is interesting since it has the best topological behavior: namely, it fails to be embedded only at the north and south pole where it has a transversal double point
There are very few results regarding the evolution of compact Lagrangian submanifolds in Cn

Summary

Introduction

At the singular time the limit surface pictures like a connect sum of a smooth Lagrangian (diffeomorphic to a Lagrangian plane) with a Whitney sphere. Such construction were later generalized to 4-dimensional Calabi–Yau manifolds, see [10]. There are very few results regarding the evolution of compact Lagrangian submanifolds in Cn. Motivate by this, we investigate the evolution of the Whitney sphere along mean curvature flow. The proof of Theorem 1.3 follows closely the ideas in [8,9] where it is shown that singularities for the mean curvature flow of monotone Lagrangian submanifolds in R4 are modeled on area minimizing cones

Preliminaries

Proof of the Theorem