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
On quantum limits on flat tori
Dmitry Jakobson
keywords: weak * limits Fourier series. resonances Laplacian flat tori Pell equation
On small gaps in the length spectrum
Dmitry Dolgopyat Dmitry Jakobson
We discuss upper and lower bounds for the size of gaps in the length spectrum of negatively curved manifolds. For manifolds with algebraic generators for the fundamental group, we establish the existence of exponential lower bounds for the gaps. On the other hand, we show that the existence of arbitrarily small gaps is topologically generic: this is established both for surfaces of constant negative curvature (Theorem 3.1) and for the space of negatively curved metrics (Theorem 4.1). While arbitrarily small gaps are topologically generic, it is plausible that the gaps are not too small for almost every metric. One result in this direction is presented in Section 5.
keywords: negatively curved manifolds Length spectrum prevalence hyperbolicity Diophantine approximations.
On the spectrum of geometric operators on Kähler manifolds
Dmitry Jakobson Alexander Strohmaier Steve Zelditch
On a compact Kähler manifold, there is a canonical action of a Lie-superalgebra on the space of differential forms. It is generated by the differentials, the Lefschetz operator, and the adjoints of these operators. We determine the asymptotic distribution of irreducible representations of this Lie-superalgebra on the eigenspaces of the Laplace--Beltrami operator. Because of the high degree of symmetry, the Laplace--Beltrami operator on forms can not be quantum ergodic. We show that, after taking these symmetries into account, quantum ergodicity holds for the Laplace--Beltrami operator and for the Spin$^\cbb$-Dirac operators if the unitary frame flow is ergodic. The assumptions for our theorem are known to be satisfied for instance for negatively curved Kähler manifolds of odd complex dimension.
keywords: Kähler manifold frame flow Dirac operator quantum ergodicity eigenfunction
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

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