Subsets of Grassmannians Preserved by Mean Curvature Flows

2002/09/16 by Wang, Mu-Tao
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.math/0209201

Abstract

Let M=Σ₁× Σ₂ be the product of two compact Riemannian manifolds of dimension n≥ 2 and two, respectively. Let Σ be the graph of a smooth map f:Σ₁↦ Σ₂, then Σ is an n-dimensional submanifold of M. Let \frak G be the Grassmannian bundle over M whose fiber at each point is the set of all n-dimensional subspaces of the tangent space of M. The Gauss map γ:Σ↦ \frakG assigns to each point x∈ Σ the tangent space of Σ at x. This article considers the mean curvature flow of Σ in M. When Σ₁ and Σ₂ are of the same non-negative curvature, we show a sub-bundle \frakS of the Grassmannian bundle is preserved along the flow, i.e. if the Gauss map of the initial submanifold Σ lies in \frakS, then the Gauss map of Σ_t at any later time t remains in \frakS. We also show that under this initial condition, the mean curvature flow remains a graph, exists for all time and converges to the graph of a constant map at infinity . As an application, we show that if f is any map from Sⁿ to S² and if at each point, the restriction of df to any two dimensional subspace is area decreasing, then f is homotopic to a constant map.

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