Grant: Discovery Grants Program - Individual

The purpose of this program is to study applications of curvature flows to convex and affine geometry. Its main part concerns flows of convex hypersurfaces by certain powers of the Gauss curvature. In essence, these flows will be studied to understand how a hypersurface evolves in time to a more optimal shape and to get significant information on these asymptotic shapes. One of the famous questions that arises within the Brunn-Minkowski-Firey theory of convex bodies generalizes the classical Minkowski problem of finding convex bodies with prescribed curvature function. In the extension of the classical problem, one seeks convex bodies with prescribed L_p curvature function, a question which generated a lot attention in view of its further applications in other areas like information theory. Past work that we did on a case of the generalized problem has led to results on the existence and uniqueness of such convex bodies via the asymptotic behavior of a curvature flow. One of the key tasks is to refine this method for the aforementioned flows to solve a class of other, more difficult, cases. There are also a number of other geometric questions in the project. They concentrate along two main directions: the study of new affine invariants of convex bodies and the question of finding sufficient conditions for two bounded convex domains in the Euclidean plane or space such that one domain contains a congruent copy of the other. Despite their apparent diversity, our objectives have as a common thread the use of geometric evolution equations. A variety of techniques coming from PDE, convex and differential geometry are necessary to approach these topics, each in itself a challenging and important problem in the field.