Nullspaces of conformally invariant operators. Applications to $\boldsymbol{Q_k}$-curvature
Yaiza Canzani A. Rod Gover Dmitry Jakobson Raphaël Ponge
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
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
Fixed frequency eigenfunction immersions and supremum norms of random waves
Yaiza Canzani Boris Hanin
A compact Riemannian manifold may be immersed into Euclidean space by using high frequency Laplace eigenfunctions. We study the geometry of the manifold viewed as a metric space endowed with the distance function from the ambient Euclidean space. As an application we give a new proof of a result of Burq-Lebeau and others on upper bounds for the sup-norms of random linear combinations of high frequency eigenfunctions.
keywords: Immersions by eigenfunctions supremum norms random waves.

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