Scalar Curvature and Q-Curvature of Random Metrics

@article{Canzani2010ScalarCA,
  title={Scalar Curvature and Q-Curvature of Random Metrics},
  author={Yaiza Canzani and Dmitry Jakobson and Igor Wigman},
  journal={The Journal of Geometric Analysis},
  year={2010},
  volume={24},
  pages={1982-2019}
}
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… 
3 Citations
THE MANIFOLD OF METRICS WITH A FIXED VOLUME FORM
We study the manifold of all metrics with the xed volume form on a compact Riemannian manifold of dimension 3. We compute the char- acteristic function for the L 2 (Ebin) distance to the reference
Critical radius and supremum of random spherical harmonics (II)
TLDR
The study of the critical radius of embeddings, via deterministic spherical harmonics, of fixed dimensional spheres into higher dimensional ones, is extended to mixed degrees, obtaining larger lower bounds on critical radii than previously found.
Gaussian measures on the of space of Riemannian metrics
We introduce Gaussian-type measures on the manifold of all metrics with a fixed volume form on a compact Riemannian manifold of dimension $$\ge $$≥3. For this random model we compute the

References

SHOWING 1-10 OF 97 REFERENCES
Constant Q-curvature metrics in arbitrary dimension
Working in a given conformal class, we prove existence of constant Q-curvature metrics on compact manifolds of arbitrary dimension under generic assumptions. The problem is equivalent to solving a
Ambient metric construction of Q-curvature in conformal and CR geometries
We give a geometric derivation of Branson's Q-curvature in terms of the ambient metric associated with conformal structures; it naturally follows from the ambient metric construction of conformally
The metric geometry of the manifold of Riemannian metrics over a closed manifold
We prove that the L2 Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold induces a metric space structure. As the L2 metric is a weak
Non-perturbative conformal quantum gravity
A quantised metric is viewed here as a probability measure on the space of (possibly degenerate) metrics on a manifold. For simplicity, attention is confined to the conformal class of a fixed
Extremal metrics for the first eigenvalue of the Laplacian in a conformal class
Let M be a compact manifold. First, we give necessary and sufficient conditions for a Riemannian metric on M to be extremal for λ 1 with respect to conformal deformations of fixed volume. In
Existence of conformal metrics with constant Q-curvature
Given a compact four dimensional manifold, we prove existence of conformal metrics with constant Q-curvature under generic assumptions. The problem amounts to solving a fourth-order nonlinear
$Q$-Curvature and Poincaré Metrics
This article presents a new definition of Branson's Q-curvature in even-dimensional conformal geometry. We derive the Q-curvature as a coefficient in the asymptotic expansion of the formal solution
Laplacian eigenvalue functionals and metric deformations on compact manifolds
Abstract In this paper, we investigate critical points of the eigenvalues of the Laplace operator considered as functionals on the space of Riemannian metrics or a conformal class of metrics on a
Manifolds of Positive Scalar Curvature: A Progress Report
In the special case n = 2, the scalar curvature is just twice the Gaussian curvature. This paper will deal with bounds on the scalar curvature, and especially, with the question of when a given
Conformal Spectrum and Harmonic maps
This paper is devoted to the study of the conformal spectrum (and more precisely the first eigenvalue) of the Laplace-Beltrami operator on a smooth connected compact Riemannian surface without
...
1
2
3
4
5
...