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 $p$ : $\lim_{t \to \infty} t^{3/2}e^{λ_0 t} p(t,x,y)=C(x,y),$ where $λ_0$ 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 $λ_0$-Martin boundary of $X$ is equal to its topological boundary. The Martin decomposition of $C(x,y)$ gives a family of measures $\{μ^{λ_0}_x \}$ on $\partial \widetilde{M}$. We show that $\{μ^{λ_0}_x \}$ is the unique family minimizing the energy or the Rayleigh quotient of Mohsen. We use the uniform Harnack inequality on the boundary $\partial X$ and the uniform three-mixing of the geodesic flow on the unit tangent bundle $SM$ for suitable Gibbs-Margulis measures.

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