The Klein quartic maximizes the multiplicity of the first positive eigenvalue of the Laplacian

Abstract

We prove that Klein quartic maximizes the multiplicity of the first positive eigenvalue of the Laplacian among all closed hyperbolic surfaces of genus $3$, with multiplicity equal to $8$. We also obtain partial results in genus $2$, where we find that the maximum multiplicity is between $3$ and $6$. Along the way, we show that for every $g\ge 2$, there exists some ${\delta }_g>0$ such that the multiplicity of any eigenvalue of the Laplacian on a closed hyperbolic surface of genus $g$ in the interval $[0,1/4+{\delta }_g]$ is at most $2g-1$ despite the fact that this interval can contain arbitrarily many eigenvalues. This extends a result of Otal to a larger interval but with a weaker bound, which nevertheless improves upon the general upper bound of Sévennec.

v1: 28 pages, 2 figures, Sage worksheets for computer-assisted proofs available under ancillary files. v2: Minor corrections made to some ancillary files and the proofs of Propositions 5.5 and 5.6. v3: Small correction made to subsection 5.1. v4: Fixed more typos. Final version to appear in Journal of Differential Geometry

Cited by in corpus (1)

• Maximal multiplicity of Laplacian eigenvalues in negatively curved surfaces