Let ℘(t, x, y) be the heat kernel on the universal cover M’ of a closed Riemannian manifold of negative curvature. We show the local limit theorem for ℘ : limt→∞ t3/2 eλ0t ℘(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)-eigenfunction which depends on x, y ∈ M’.
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