The Heat kernel on Asymptotic Hyperbolic Manifolds

We will show the upper bound of the heat kernel, via the corresponding resolvent on AH spaces due to Mazzeo-Melrose, Melrose-Sa Barreto-Vasy. This result generalises the upper bound on hyperbolic spaces by Davies-Mandouvalos. In the mean time, it yields the $L^p$ boundedness of the Riesz transform. This is a joint work with A. Hassell.