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Maximally degenerate laplacians

Steven Zelditch (1996)

Annales de l'institut Fourier

The Laplacian Δ g of a compact Riemannian manifold ( M , g ) is called maximally degenerate if its eigenvalue multiplicity function m g ( k ) is of maximal growth among metrics of the same dimension and volume. Canonical spheres ( S n , can ) and CROSSes are MD, and one asks if they are the only examples. We show that a MD metric must be at least a Zoll metric with just one distinct eigenvalue in each cluster, and hence with all band invariants equal to zero. The principal band invariant is then calculated in terms of geodesic...

Measure-geometric Laplacians for partially atomic measures

Marc Kesseböhmer, Tony Samuel, Hendrik Weyer (2020)

Commentationes Mathematicae Universitatis Carolinae

Motivated by the fundamental theorem of calculus, and based on the works of W. Feller as well as M. Kac and M. G. Kreĭn, given an atomless Borel probability measure η supported on a compact subset of ℝ U. Freiberg and M. Zähle introduced a measure-geometric approach to define a first order differential operator ∇ η and a second order differential operator Δ η , with respect to η . We generalize this approach to measures of the form η : = ν + δ , where ν is non-atomic and δ is finitely supported. We determine analytic...

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