A geometric stability inequality and applications to the stability of geometric flows

In this talk I will discuss some results concerning geometric flows. In particular, we will focus on flows with a volume constraint and whose motion is depending on their mean curvature, which (formally) arise as gradient flows for the perimeter functional.
After an introduction on the topic, aimed at a general audience, we will discuss a novel geometric inequality, which takes the form of a quantitative Alexandrov theorem, in the periodic setting. We will then show how to use this inequality to prove global existence and to characterize the asymptotic behaviour for some instances of volume-preserving geometric flows. Our results apply to the volume-preserving mean curvature flow, the surface diffusion flow and the Mullins-Sekerka flow.
This work is based on a collaboration with Anna Kubin (TU Wien), Andra Kubin (University of Jyväskylä) and Antonia Diana (Sapienza University).