Differential Geometry Seminar
This is a joint work with Pedro Valentin De Jesus. A well known result of Gage states that mean curvature flow (MCF) improves the isoperimetric quotient of a convex 2 body and evolves it into a round point. However, the definition of MCF requires certain local regularity. For example, it is tricky to define MCF for a triangle or a square. In this work, we propose a global deformation of a convex 2-dimensional region into a round disc, improving the isoperimetric quotient along the way. We explain the case of polygons in detail. We then discuss several variations of the scheme, posing some additional constraints. As application, we propose some sharp Isoperimetric inequalities under various constraints, improving some almost century old results of Benneson and others. High dimensional extensions are also possible.
In this talk we will just outline our approach. Details will be discussed in future talks.