Consider the heat kernel on the universal cover of a closed Riemannian manifold of negative sectional curvature. We show the local limit theorem for ℘:
where is the bottom of the spectrum of the geometric Laplacian and is a positive -harmonic function which depends on . We also show that the -Martin boundary of is equal to its topological boundary. The Martin decomposition of gives a family of measures on . We show that is a family minimizing the energy or Mohsen’s Rayleigh quotient. We apply the uniform Harnack inequality on the boundary and the uniform three-mixing of the geodesic flow on the unit tangent bundle for suitable Gibbs–Margulis measures.
"Local limit theorem in negative curvature." Duke Math. J. 170 (8) 1585 - 1681, 1 June 2021. https://doi.org/10.1215/00127094-2020-0069