几何分析系列报告之08【 Gioacchino Antonelli 】

报告摘要：In this talk, I will explore how a bound from below on the Ricci tensor affects the isoperimetric properties of a space. The setting I will be dealing with is the one of RCD(K,N) metric measure spaces, i.e., metric measure spaces with a synthetic notion of Ricci curvature bounded from below by K and dimension bounded above by N. The results discussed are new in the non compact smooth Riemannian setting, and I will highlight how the non-smooth geometry naturally comes into play even when dealing with the smooth case. At the beginning of the seminar, I will give a brief introduction to the RCD theory, I will recall the notion of perimeter of a set in a metric measure space, and the formulation of the isoperimetric problem. I will then describe an asymptotic mass decomposition theorem for minimizing (for the perimeter) sequences of finite perimeter sets with fixed volume. It might happen that in the minimization process in the non compact setting part of the mass escapes at infinity in a metric measure space that could happen to be non smooth. The notion of convergence used here is the pointed measured Gromov–Hausdorff one. I will then show how this asymptotic mass decomposition can give new information on the fine properties of the isoperimetric profile. In particular, I will prove a sharp second order differential inequality for the isoperimetric profile on N-dimensional RCD(K,N) spaces with a uniform lower bound on the volumes of unit balls. One of the key tools needed is a new (weak) notion of having constant mean curvature tailored for isoperimetric boundaries in metric measure spaces. Finally, I will discuss several consequence of the sharp differential inequality. This seminar is self-contained. People who are interested in these topics might also follow the companion seminar by Marco Pozzetta held on April 17th. This talk is based on several joint collaborations with E. Bru` e, M. Fogagnolo, S. Nardulli, E. Pasqualetto, M. Pozzetta, D. Semola, I. Y. Violo.