Differential Geometry Seminar: Continuous quantities with respect to measured Gromov-Hausdorff convergence

Seminar | January 29 | 2:10-3 p.m. | 939 Evans Hall

 Shouhei Honda, Tohoku University

 Department of Mathematics

One of main purposes in the convergence theory (with uniform Ricci bounds from below) is to find geometric/analytic quantities which are continuous with respect to measured Gromov-Hausdorff convergence. The diameter is a trivial geometric example. On the other hand the \(k^{th}\) eigenvalue of the Laplacian is a nontrivial analytic example for all \(k\), which was proven by Cheeger-Colding. In this talk we provide a new nontrivial geometric example, so-called Cheeger's isoperimetric constant. In the proof, BV-functions and \(L^1\)-Bakry-Emery estimate play key roles. Moreover we give a deep relationship between these three continuous quantities via PDEs. This is a joint work with L. Ambrosio.