Local Limit Theorem in negative curvature

Consider the heat kernel $p(t,x,y)$ on the universal cover $X$ of a Riemannian manifold $M$ of negative curvature. We show the local limit theorem for $p$ : $$\underset{t\to \mathrm{\infty }}{\lim }{t}^{3\mathrm{/}2}{e}^{{\lambda }_{0}t}p(t,x,y)=C(x,y),$$ where ${\lambda }_0$ is the bottom of the spectrum of the geometric Laplacian and $C(x,y)$ is a positive function which depends on $x,y\in X$. We also show that the ${\lambda }_0$-Martin boundary of $X$ is equal to its topological boundary. The Martin decomposition of $C(x,y)$ gives a family of measures $\{{\mu }_x^{{\lambda }_0}\}$ on $\mathrm{\partial }\tilde{M}$. We show that $\{{\mu }_x^{{\lambda }_0}\}$ is the unique family minimizing the energy or the Rayleigh quotient of Mohsen. We use the uniform Harnack inequality on the boundary $\mathrm{\partial }X$ and the uniform three-mixing of the geodesic flow on the unit tangent bundle $SM$ for suitable Gibbs-Margulis measures.