Nullspaces of conformally invariant operators. Applications to $\boldsymbol{Q_k}$-curvature
Yaiza Canzani A. Rod Gover Dmitry Jakobson Raphaël Ponge
Electronic Research Announcements 2013, 20(0): 43-50 doi: 10.3934/era.2013.20.43
We study conformal invariants that arise from functions in the nullspace of conformally covariant differential operators. The invariants include nodal sets and the topology of nodal domains of eigenfunctions in the kernel of GJMS operators. We establish that on any manifold of dimension $n\geq 3$, there exist many metrics for which our invariants are nontrivial. We discuss new applications to curvature prescription problems.
keywords: conformal geometry Spectral geometry Qk-curvature. nodal sets
Scalar curvature and $Q$-curvature of random metrics
Yaiza Canzani Dmitry Jakobson Igor Wigman
Electronic Research Announcements 2010, 17(0): 43-56 doi: 10.3934/era.2010.17.43
We study Gauss curvature for random Riemannian metrics on a compact surface, lying in a fixed conformal class; our questions are motivated by comparison geometry. We next consider analogous questions for the scalar curvature in dimension $n>2$, and for the $Q$-curvature of random Riemannian metrics.
keywords: excursion probability $Q$-curvature conformally covariant operators. Laplacian conformal class Comparison geometry Gaussian random fields scalar curvature

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