Geodesic
The heat equation on noncompact Riemannian manifolds
Sbornik. Mathematics, Tome 72 (1992) no. 1, pp. 47-77

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The behavior of the Green function $G(x,y,t)$ of the Cauchy problem for the heat equation on a connected, noncompact, complete Riemannian manifold is investigated. For manifolds with boundary it is assumed that the Green function satisfies a Neumann condition on the boundary. 
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     author = {A. A. Grigor'yan},
     title = {The heat equation on noncompact {Riemannian} manifolds},
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A. A. Grigor'yan. The heat equation on noncompact Riemannian manifolds. Sbornik. Mathematics, Tome 72 (1992) no. 1, pp. 47-77. http://geodesic.mathdoc.fr/item/SM_1992_72_1_a2/