Mathematics

Volume preserving mean curvature flow is the evolution of a compact hypersurface with speed given by the mean curvature plus a nonlocal term which keeps the enclosed volume constant. The additional term induces some difficulties in the analysis, such as the failure of the avoidance principle. On the other hand, the isoperimetric ratio of the region enclosed by the hypersurface is decreasing in time, in contrast to the standard mean curvature flow. The monotonicity of the isoperimetric ratio turns out to be a powerful tool for studying the long time behaviour of convex hypersurfaces and to prove convergence to a spherical profile. We present some recent extensions of this approach to other classes of volume preserving geometric flows, driven by general powers of the mean curvature or by fractional mean curvature (in collaboration with E. Cinti and E. Valdinoci) and to capillary hypersurfaces with a prescribed angle condition on the boundary (in collaboration with L. Weng). Volume preserving mean curvature flow has also found applications in General Relativity as a tool to construct a foliation by constant mean curvature surfaces of the outer region of a three-manifold modelling an isolated gravitating system. This approach was originally introduced by Huisken and Yau in the context of asymptotically Schwarzschild spaces and we present here a generalization to a larger class of asymptotically flat manifolds with weaker decay assumptions (in collaboration with J. Tenan).