A regularity theorem for graphic spacelike mean curvature flows

Abstract

For mean curvature flows in Euclidean spaces, Brian White proved a regularity theorem which gives C 2; estimates in regions of spacetime where the Gaussian density is close enough to 1. This is proved by applying Huisken’s monotonicity formula. Here we will consider mean curvature flows in semiEuclidean spaces, where each spatial slice is an m-dimensional graph in R mCn n satisfying a gradient bound stronger than the spacelike condition. By defining a suitable quantity to replace the Gaussian density ratio, we prove monotonicity theorems similar to Huisken’s and use them to prove a regularity theorem similar to White’s.