q-parabolicity of stratified pseudomanifolds and other singular spaces

Abstract

The main result of this paper is a sufficient condition to have a compact Thom–Mather stratified pseudomanifold endowed with a \(\hat{c}\)-iterated edge metric on its regular part q-parabolic. Moreover, besides stratified pseudomanifolds, the q-parabolicity of other classes of singular spaces, such as compact complex Hermitian spaces, is investigated.

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Notes

  1. 1.

    \(\mathrm {e}^{-tH_g}\) has a smooth integral kernel which satisfies \(\int _M \mathrm {e}^{-tH_g}(x,y)\mathrm{d}\mu _g(y)\le 1\) for all \(t>0\), \(x\in M\), so that one can define \(\mathrm {e}^{-tH_g}f(x)\) for bounded functions f on M by \(\mathrm {e}^{-tH_g}f(x):=\int _M \mathrm {e}^{-tH_g}(x,y)f(y)\mathrm{d}\mu _g.\)

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Acknowledgments

The authors wish to thank the anonymous referee for his valuable comments that in particular led to the current formulation of Proposition 4.5. The second named author (B.G.) would like to thank Stefano Pigola for many motivating discussions. Both authors have been financially supported by SFB 647: Raum-Zeit-Materie.

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Correspondence to Francesco Bei.

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Bei, F., Güneysu, B. q-parabolicity of stratified pseudomanifolds and other singular spaces. Ann Glob Anal Geom 51, 267–286 (2017). https://doi.org/10.1007/s10455-016-9534-0

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Keywords

  • q-parabolicity
  • Stochastic completeness
  • Heat kernel
  • Stratified pseudomanifold
  • Iterated edge metric
  • Hermitian complex space