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 $g\geq 2$, 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 S\'evennec.