18 may 2016 -- 17:30   [open in google calendar]

sala seminari, dipartimento di matematica di Pisa

Abstract.

The two formulae of the heat kernel on $\mathbb{R}^n$ and on the hyperbolic space$\mathbb{H}^n$ give a first example of the fact that the geometry of a space has an incidence on the heat kernel. On general manifolds, heat kernels are almost never expressed by such explicit formulae. However, under some hypothesis on the geometry of the manifold, nice upper or lower bounds of the heat kernel can be obtained. In an article on 1986, P. Li used such bounds to obtain a first result concerning the asymptotic behavior of the heat kernel on manifolds with nonnegative Ricci curvature and maximal volume growth. In 2013, Xu obtained a similar result removing the maximal volume growth hypothesis. The object of the seminar will be to present Li's and Xu's works, stressing the use by Xu of Cheeger-Colding's theory.