UBC Mathematics Colloquium

Bill Minicozzi
(Johns Hopkins University, USA)

Generic singularities of mean curvature flow

Fri., Oct. 2, 2009, 3:00pm, MATX 1100

In mean curvature flow (or MCF), a surface evolves to minimize its surface area as quickly as possible.  One of the challenges of MCF is that the flow starting from a closed surface (like a sphere) always becomes singular and one of the most important problems is understanding these singularities.  The simplest example comes from a round sphere, which evolves by staying round but having the radius shrink until it hits zero and then just disappears.  Matt Grayson proved that this is the only type of singularity that occurs for simple closed curves in the plane.  However, many other examples were discovered in higher dimensions (most of them by applied mathematicians doing numerical simulations).  I will describe recent work with Toby Colding, MIT, where we classified the generic singularities of MCF of closed embedded hypersurfaces.  The thrust of our result is that, in all dimensions, every singularity other than shrinking spheres and cylinders can be perturbed away.  

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