Local limit theorem in negative curvature

Consider the heat kernel ℘(t,x,y) on universal cover M˜ of a closed Riemannian manifold negative sectional curvature. We show local limit theorem for ℘: limt→∞t3∕2eλ0t℘(t,x,y)=C(x,y), where λ0 is bottom spectrum geometric Laplacian and C(x,y) positive λ0-harmonic function which depends x,y∈M˜. also that λ0-Martin boundary equal to its topological boundary. The Martin decomposition gives family measures {μxλ0} ∂M˜. {μ xλ0} minimizing energy or Mohsen’s Rayleigh quotient. apply uniform Harnack inequality ∂M˜ three-mixing geodesic flow unit tangent bundle SM suitable Gibbs–Margulis measures.