Harmonic morphisms and subharmonic functions. Choi, Gundon, Yun, Gabjin (2005) International Journal of Mathematics and Mathematical Sciences
Heat kernel on manifolds with ends Alexander Grigor’yan, Laurent Saloff-Coste (2009) Annales de l’institut Fourier We prove two-sided estimates of heat kernels on non-parabolic Riemannian manifolds with ends, assuming that the heat kernel on each end separately satisfies the Li-Yau estimate.