Title: Nonlinear Parabolic Equations and Isoperimetric Type Inequalities
Speaker: Pengfei Guan (McGill University)
Abstract: We discuss applications of nonlinear parabolic equations in differential geometry. Isoperimetric type geometric inequalities are of interest not only in geometry, also in analysis and mathematical physics. The equality cases are often attained by optimal shape objects in geometry. Recently, there have been some new developments using evolution equations/geometric flows. For example, to establish isoperimetric type inequality for two given geometric quantities, one searches for an evolution equation such as that along the flow, one of the quantities is invariant and another is monotone. The convergence of the flow to an equilibrium solution would yield the desired inequality. The flows are fully nonlinear versions of modified mean curvature or inverse mean curvature flows. The main focus is the longtime existence and convergence of the corresponding flows. The talk is for a general audience in mathematics.
