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,α 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 ‫ޒ‬ m+n 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.

The maximal graph Dirichlet problem in semi-Euclidean spaces

“…Such a result (but for minimal graphs) is the goal of [10], where the proof uses a gradient estimate and White's regularity theorem (see [11]) for mean curvature flows. For spacelike mean curvature flows in semi-Euclidean spaces, we have a similar gradient estimate (see Proposition 2) and a version of White's theorem (see [9]), so it makes sense to attempt a similar proof. In [10], White's theorem is used to get C 2,α estimates to prove long time existence.…”

Lagrangian mean curvature flow with boundary

“…Although no work has been done on Lagrangian mean curvature flow with boundary conditions (other than curve-shortening flow), an alternative boundary condition has been studied by Butscher [2][3] for the related elliptic case of special Lagrangians with boundary on a codimension 2 symplectic submanifold. Boundary conditions for codimension 1 mean curvature flow have been considered in a variety of contexts, for example by Ecker [4], Priwitzer [24] and Thorpe [34] in the Dirichlet case, by Buckland [1], Edelen [6][7], Huisken [13], Lambert [16][17], Lira-Wanderley [19], Stahl [31][32] and Wheeler [37][38] in the Neumann case, and by Wheeler-Wheeler [36] in a mixed Dirichlet Neumann case.…”

Mean curvature flow of compact spacelike submanifolds in higher codimension

“…Mean curvature flow of spacelike hypersurfaces in indefinite spaces has been studied previously, for example [6] [7] [20], as has higher codimension mean curvature flow in definite spaces [2] [5] [14] [18] [21].…”