# Scalar Curvature and *Q*-Curvature of Random Metrics

## Abstract

We define a family of probability measures on the set of Riemannian metrics lying in a fixed conformal class, induced by Gaussian probability measures on the (logarithms of) conformal factors. We control the smoothness of the resulting metric by adjusting the decay rate of the variance of the random Fourier coefficients of the conformal factor. On a compact surface, we evaluate the probability of the set of metrics with non-vanishing Gauss curvature, lying in a fixed conformal class. On higher-dimensional manifolds, we estimate the probability of the set of metrics with non-vanishing scalar curvature (or *Q*-curvature), lying in a fixed conformal class.

## Keywords

Conformal class Metrics with non-vanishing scalar curvature*Q*-curvature Gaussian random fields Excursion probability Laplacian Conformally covariant operators

## Mathematics Subject Classification

53A30 53C21 58J50 58D17 58D20 60G60## Notes

### Acknowledgements

The authors would like to thank R. Adler, P. Guan, V. Jaksic, N. Kamran, S. Molchanov, I. Polterovich, G. Samorodnitsky, B. Shiffman, J. Taylor, J. Toth, K. Worsley, and S. Zelditch for stimulating discussions about this problem. The authors are also grateful to the referee for useful remarks. The authors would like to thank for their hospitality the organizers of the following conferences, where part of this research was conducted: “Random Functions, Random Surfaces and Interfaces” at CRM (January, 2009); “Random Fields and Stochastic Geometry” at Banff International Research Station (February, 2009). In addition, D.J. would like to thank the organizers of the program “Selected topics in spectral theory” at Erwin Shrödinger Institute in Vienna (May 2009), as well as the organizers of the conference “Topological Complexity of Random Sets” at American Institute of Mathematics in Palo Alto (August 2009).

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