Small eigenvalues of surfaces
Werner Ballmann, Henrik Matthiesen, Sugata Mondal
TL;DR
This paper proves a bound on the number of small eigenvalues of the Laplacian for Riemannian metrics on closed surfaces with negative Euler characteristic, linking spectral properties to topological invariants.
Contribution
It establishes a sharp upper bound on the count of small eigenvalues based on the Euler characteristic of the surface.
Findings
Number of small eigenvalues ≤ -χ(S) for surfaces with χ(S)<0
Provides a spectral-topological relationship for closed surfaces
Advances understanding of Laplacian spectra on Riemannian surfaces
Abstract
We show that the Laplacian of a Riemannian metric on a closed surface S with Euler characteristic \chi(S) < 0 has at most -\chi(S) small eigenvalues.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Nonlinear Partial Differential Equations · Analytic and geometric function theory