The Existence of a Weak Solution to Volume Preserving Mean Curvature Flow in Higher Dimensions

In this paper, we construct a family of integral varifolds, which is a global weak solution to the volume preserving mean curvature flow in the sense of L-2-flow. This flow is also a distributional BV-solution for a short time, when the perimeter of the initial data is sufficiently close to that of a ball with the same volume. To construct the flow, we use the Allen-Cahn equation with a non-local term motivated by studies of Mugnai, Seis, and Spadaro, and Kim and Kwon. We prove the convergence of the solution for the Allen-Cahn equation to the family of integral varifolds with only natural assumptions for the initial data.

Automated summary

In this paper, a family of integral varifolds is constructed as a global weak solution to the volume preserving mean curvature flow in the sense of L-2-flow. The flow is also a distributional BV-solution for a short time when the perimeter of the initial data is close enough to that of a ball with the same volume. The construction of the flow involves the use of the Allen-Cahn equation with a non-local term motivated by previous studies.