Event № 184
Mean curvature flow is a geometric heat equation for hyper-surfaces, which is the gradient flow of the surface area functional. The flow typically becomes singular at finite time, after which it can be extended by an object called the "level set flow". In general, the level set flow is not that well behaved, but in the important mean convex case, where the initial hypersurface is a boundary of a domain which starts moving inward, a beautiful regularity and structure theory was developed in the last 20 years by Brian White. While parts of this theory work in full generality, parts were only known to hold in either the Euclidean setting or in low dimensions.
We prove two new estimates for the level set flow of mean convex domains in general Riemannian manifolds. Our estimates give control - exponential in time - for the infimum of the mean curvature, and the ratio between the norm of the second fundamental form and the mean curvature. In particular, the estimates remove the above mentioned stumbling block that has been left after the work of White and thus allow us to extend the structure theory for weak mean convex level set flow to general ambient manifolds of arbitrary dimension. While the setting and motivation of the work is geometric, almost the entire labor turned out to be analytic. For instance, what is readily seen to be the main obstacle in generalizing the result from the Euclidean to the non-Euclidean setting is, in fact, the lack of an a-priori C^0 estimate for solutions to a certain family of elliptic PDEs. Although completely decoupling the analysis from the geometry of the problem would be misleading, the talk will highlight the analytic aspects of the work, and should be accessible to everyone doing analysis. This is a joint work with Robert Haslhofer.