Local limit theorem in negative curvature

Abstract

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