Heat kernel estimates on connected sums of parabolic manifolds

2016/08/04 by Alexander Grigor'yan, Grigor'yan, Alexander, Satoshi Ishiwata +3
Mathematics · #35K08 (Primary) #58J35 (Secondary) #58J65 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:35K08 #msc:58J35 #msc:58J65

paper · pdf · doi:10.48550/arxiv.1608.01596

38 pages, 7 figures

arxiv created 2016/08/04 · arxiv updated 2016/08/05

Abstract

We obtain matching two sided estimates of the heat kernel on a connected sum of parabolic manifolds, each of them satisfying the Li-Yau estimate. The key result is the on-diagonal upper bound of the heat kernel at a central point. Contrary to the nonparabolic case (which was settled in [15]), the on-diagonal behavior of the heat kernel in our case is determined by the end with the maximal volume growth function. As examples, we give explicit heat kernel bounds on the connected sums R²#R² and R¹#R² where R¹ = R₊\timesS¹.

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