Gaussian upper bounds for the heat kernel on arbitrary manifolds

The history of the heat kernel Gaussian estimates started with the works of Nash [25] and Aronson [2] where the double-sided Gaussian estimates were obtained for the heat kernel of a uniformly parabolic equation in IR in a divergence form (see also [15] for improvement of the original Nash’s argument and [26] for a consistent account of the Aronson’s results and related topics). In particular, the Aronson’s upper bound for the case of time-independent coefficients which is of interest for us reads as follows:

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