Abstract
Let (X, g) be a product cone with the metric \(g=dr^2+r^2h\), where \(X=C(Y)=(0,\infty )_r\times Y\) and the cross section Y is a \((n-1)\)-dimensional closed Riemannian manifold (Y, h). We study the upper boundedness of heat kernel associated with the operator \({\mathcal {L}}_V=-\Delta _g+V_0 r^{-2}\), where \(-\Delta _g\) is the positive Friedrichs extension Laplacian on X and \(V=V_0(y) r^{-2}\) and \(V_0\in {\mathcal {C}}^\infty (Y)\) is a real function such that the operator \(-\Delta _h+V_0+(n-2)^2/4\) is a strictly positive operator on \(L^2(Y)\). The new ingredient of the proof is the Hadamard parametrix and finite propagation speed of wave operator on Y.
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Acknowledgements
The authors would like to thank Hongquan Li for his discussions and comments that improved the exposition. JZ acknowledges support from National key R &D program of China: No. 2022YFA1005700 and National Natural Science Foundation of China (12171031,11831004), XH acknowledges support from an AMS-Simons travel grant.
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Huang, X., Zhang, J. Heat Kernel Estimate in a Conical Singular Space. J Geom Anal 33, 284 (2023). https://doi.org/10.1007/s12220-023-01348-0
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DOI: https://doi.org/10.1007/s12220-023-01348-0