On Doyle-Grigor’yan criterion for non-parabolicity

Bessa, G.; Gimeno i Garcia, Vicent; Pessoa, Leandro; Setti, Alberto

doi:10.1090/proc/16804

On Doyle-Grigor’yan criterion for non-parabolicity
HTML articles powered by AMS MathViewer

by G. Pacelli Bessa, Vicent Gimeno i Garcia, Leandro F. Pessoa and Alberto G. Setti

Proc. Amer. Math. Soc.

DOI: https://doi.org/10.1090/proc/16804

Published electronically: April 11, 2024

HTML | PDF | Request permission

Abstract:

In this short note we show that Doyle-Grigor’yan criterion for non-parabolicity is not necessary in dimension greater than or equal to four. This gives an negative answer to Problem # 1 of Grigor’yan [Bull. Amer. Math. Soc. (N.S) 36 (1999). pp. 135–249] in this dimensional range.

References

Isaac Chavel and Leon Karp, Large time behavior of the heat kernel: the parabolic $\lambda$-potential alternative, Comment. Math. Helv. 66 (1991), no. 4, 541–556. MR 1129796, DOI 10.1007/BF02566664
Leung-Fu Cheung and Pui-Fai Leung, Eigenvalue estimates for submanifolds with bounded mean curvature in the hyperbolic space, Math. Z. 236 (2001), no. 3, 525–530. MR 1821303, DOI 10.1007/PL00004840
Peter G. Doyle, On deciding whether a surface is parabolic or hyperbolic, Geometry of random motion (Ithaca, N.Y., 1987) Contemp. Math., vol. 73, Amer. Math. Soc., Providence, RI, 1988, pp. 41–48. MR 954627, DOI 10.1090/conm/073/954627
Alexander Grigor′yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 2, 135–249. MR 1659871, DOI 10.1090/S0273-0979-99-00776-4
Peter Li, Large time behavior of the heat equation on complete manifolds with nonnegative Ricci curvature, Ann. of Math. (2) 124 (1986), no. 1, 1–21. MR 847950, DOI 10.2307/1971385