Consider the heat kernel $\wp (t,x,y)$ on the universal cover $\tilde{M}$ of a closed Riemannian manifold of negative sectional curvature. We show the local limit theorem for ℘:

$$\underset{t\to \mathrm{\infty }}{lim}{t}^{3∕2}{e}^{{\mathit{\lambda }}_{0}t}\wp (t,x,y)=C(x,y),$$

where ${\mathit{\lambda }}_0$ is the bottom of the spectrum of the geometric Laplacian and $C(x,y)$ is a positive ${\mathit{\lambda }}_0$-harmonic function which depends on $x,y\in \tilde{M}$. We also show that the ${\mathit{\lambda }}_0$-Martin boundary of $\tilde{M}$ is equal to its topological boundary. The Martin decomposition of $C(x,y)$ gives a family of measures $\{{\mathrm{\mu }}_x^{{\mathit{\lambda }}_0}\}$ on $\partial \tilde{M}$. We show that $\{{\mathrm{\mu }}_x^{{\mathit{\lambda }}_0}\}$ is a family minimizing the energy or Mohsen’s Rayleigh quotient. We apply the uniform Harnack inequality on the boundary $\partial \tilde{M}$ and the uniform three-mixing of the geodesic flow on the unit tangent bundle $SM$ for suitable Gibbs–Margulis measures.