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An application of curvature flows to convex geometry

Alina Stancu
University of Massachusetts Lowell

Thursday, October 20, 2005
L01 Carson Hall, 4 pm
Tea 3:30 pm, Math Lounge

Abstract: Curvature flows are characterized by the deformation of geometric objects such as metrics, mappings, or submanifolds by geometric quantities depending on curvature. They can be used to understand some canonical or optimal states which are the states preferred by the evolving geometric object. For example, closed convex hypersurfaces in R^{n+1} evolving under the affine curvature flow tend to become ellipsoids. We will use this flow to answer, under some assumptions, a seemingly unrelated question known as the floating body problem.\par Let K \subset R^{n+1} be a convex body and let \delta >0 a sufficiently small real number. We call the floating body associated to K, denoted K_{\delta}, the intersection of all half-spaces in R^{n+1} whose complements cut from K a cap of volume \delta. Conjectured in the 90's, the floating body problem states that K is homothetic to K_{\delta} if and only if K is an ellipsoid.

This talk will be accessible to undergraduates.