Heat kernel estimates on connected sums of parabolic manifolds (Q1746488)

The paper under review completes the results from [\textit{A. Grigor'yan} and \textit{L. Saloff-Coste}, Ann. Inst. Fourier 59, No. 5, 1917--1997 (2009; Zbl 1239.58016)] by proving 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 non-parabolic case, the on-diagonal behavior of the heat kernel in the case considered is determined by the end with the maximal volume growth function. The off-diagonal estimates for the heat kernels are derived by using the gluing techniques from [loc. cit.]. As application of the results obtained, the authors give explicit heat kernel bounds on the connected sums \(\mathbb{R}^2\#\mathbb{R}^2\) and \(\mathcal{R}^1\#\mathbb{R}^2\) with \(\mathcal{R}^1=\mathbb{R}_+\times\mathbb{S}^1.\)