9/29/2006

Maria Calle
Courant Institute

Ancient solutions for mean curvature flow

In the first part of the talk, I'll introduce mean  curvature flow. A family of surfaces in R3 (or, in general, k- submanifolds in Rn) is said to move by mean curvature flow if its  movement satisfies a particular parabolic PDE. This evolution follows  the steepest descent direction for the area, that is, the surfaces  decrease their area at the fastest possible rate. I present some  basic facts about mean curvature flow solutions, such as a mean value  inequality and the definition of density at a point.

After that, I'll present a result about ancient solutions. An ancient  solution for mean curvature flow is a solution defined for all times  t<0. I give a bound on the dimension of the ambient space of an  ancient solution, depending on a bound on the density of the evolving  submanifold.