9 oct 2024 -- 15:00   [open in google calendar]

Scuola Normale Superiore, Aula Bianchi Scienze

Abstract.

The $L^p$ version of the Minkowski problem was introduced by Lutwak, an extension of the classical Minkowski problem in convex geometry. We discuss the Gauss curvature type flow approach to the problem. The Gauss curvature flow was introduced by Firey in 1974, the flow and its variations have been extensively studied, through works by K.S. Chou, R. Hamilton, B. Chow, B. Andrews, Guan-Ni, Brendle-Choi-Daskalopoulos, and many others. The convergence in general dimension was proved by Guan-Ni and Andrews-Guan-Ni, the most important uniqueness problm was resolved by Brendle-Choi-Daskalopoulos in recent years. An entropy type quantity and its estimates play a crucial role in the work of Andrews-Guan-Ni. This entropy is also related to the $L^p$-Minkowski problem, as its solution can be formulated as a critical point of entropy functional with constrained volume, a problem of calculus of variations. The anisotropic flow by power of Gauss curvature can serve as a natural path for this problem of calculus of variations. We will also some open problems related to the Christoffel-Minkowski type problem.