Local limit theorem and Martin boundary of hyperbolic manifolds

Using dynamics of the geodesic flow, we show that the heat kernel $p(t,x,y)$ on the universal cover of a closed manifold of negative curvature is asymptotically of the form $$e^{-\lambda t} t^{-3/2} C(x,y)$$, where $\lambda$ is the bottom of the spectrum of the Laplacian and $C(x,y)$ is a positive constant depending only on $x$ and $y$. We also show that the $\lambda$-Martin boundary coincides with its Gromov boundary.

We will explain how to use the uniform Harnack inequality and the uniform two-mixing of the geodesic flow for suitable Gibbs-Margulis measures. This is a joint work with F. Ledrappier.