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### Open Access Journals

ERA-MS

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.

ERA-MS

JMD

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.

JMD

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.

ERA-MS

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.

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