Hypersurface flows and isoperimetric type problems

Author: Pengfei Guan (Montreal)

Abstract: We consider flow approach for isoperimetric type problems associated with general geometric quantities. The isoperimetric problem for two geometric functionals can be viewed as a Lagrange multiplier problem in calculus variations: finding "minimum" of one functional under the constraint of the other functional fixed. One seeks a "good path" descending to the "optimal solutions". In the case of classical isoperimetric inequality, it naturally leads to a mean curvature type flow. In general, the approach yields some new curvature flows. We discuss the longtime existence and convergence of these flows and new geometric inequalities as consequences. We will also discuss some unsolved open regularity problems associated to these flows.