A discovery tour in random Riemannian geometry
Dello Schiavo L, Kopfer E, Sturm KT. 2024. A discovery tour in random Riemannian geometry. Potential Analysis.
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Author
Dello Schiavo, LorenzoISTA 
;
Kopfer, Eva;
Sturm, Karl Theodor
;
Kopfer, Eva;
Sturm, Karl TheodorDepartment
Abstract
We study random perturbations of a Riemannian manifold (M, g) by means of so-called
Fractional Gaussian Fields, which are defined intrinsically by the given manifold. The fields
h• : ω → hω will act on the manifold via the conformal transformation g → gω := e2hω g.
Our focus will be on the regular case with Hurst parameter H > 0, the critical case H = 0
being the celebrated Liouville geometry in two dimensions. We want to understand how basic
geometric and functional-analytic quantities like diameter, volume, heat kernel, Brownian
motion, spectral bound, or spectral gap change under the influence of the noise. And if so, is
it possible to quantify these dependencies in terms of key parameters of the noise? Another
goal is to define and analyze in detail the Fractional Gaussian Fields on a general Riemannian
manifold, a fascinating object of independent interest.
Publishing Year
Date Published
2024-01-26
Journal Title
Potential Analysis
Acknowledgement
The authors would like to thank Matthias Erbar and Ronan Herry for valuable discussions on this project. They are also grateful to Nathanaël Berestycki, and Fabrice Baudoin for respectively pointing out the references [7], and [6, 24], and to Julien Fageot and Thomas Letendre for pointing out a mistake in a previous version of the proof of Proposition 3.10. The authors feel very much indebted to an anonymous reviewer for his/her careful reading and the many valuable suggestions that have significantly contributed to the improvement of the paper. L.D.S. gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft through CRC 1060 as well as through SPP 2265, and by the Austrian Science Fund (FWF) grant F65 at Institute of Science and Technology Austria. This research was funded in whole or in part by the Austrian Science Fund (FWF) ESPRIT 208. For the purpose of open access, the authors have applied a CC BY public copyright licence to any Author Accepted Manuscript version arising from this submission. E.K. and K.-T.S. gratefully acknowledge funding by the Deutsche Forschungsgemeinschaft through the Hausdorff Center for Mathematics and through CRC 1060 as well as through SPP 2265.
Open Access funding enabled and organized by Projekt DEAL.
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Cite this
Dello Schiavo L, Kopfer E, Sturm KT. A discovery tour in random Riemannian geometry. Potential Analysis. 2024. doi:10.1007/s11118-023-10118-0
Dello Schiavo, L., Kopfer, E., & Sturm, K. T. (2024). A discovery tour in random Riemannian geometry. Potential Analysis. Springer Nature. https://doi.org/10.1007/s11118-023-10118-0
Dello Schiavo, Lorenzo, Eva Kopfer, and Karl Theodor Sturm. “A Discovery Tour in Random Riemannian Geometry.” Potential Analysis. Springer Nature, 2024. https://doi.org/10.1007/s11118-023-10118-0.
L. Dello Schiavo, E. Kopfer, and K. T. Sturm, “A discovery tour in random Riemannian geometry,” Potential Analysis. Springer Nature, 2024.
Dello Schiavo L, Kopfer E, Sturm KT. 2024. A discovery tour in random Riemannian geometry. Potential Analysis.
Dello Schiavo, Lorenzo, et al. “A Discovery Tour in Random Riemannian Geometry.” Potential Analysis, Springer Nature, 2024, doi:10.1007/s11118-023-10118-0.
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