The Bolza surface and Klein quartic extremal bass-note conjectures

A closed hyperbolic surface has a bass note given by its first positive Laplace eigenvalue. The Bolza surface is the genus-$2$ closed hyperbolic surface, and the Klein quartic is the genus-$3$ closed hyperbolic surface.

Bolza–Klein bass-note conjecture. (a) The Bolza surface uniquely maximizes the bass note among closed and orientable hyperbolic surfaces. (b) The Klein quartic uniquely maximizes the bass note among closed and orientable hyperbolic surfaces of genus $3$.

Numerical estimates and conformal-bootstrap upper bounds are very close to the bass notes of the Bolza surface and Klein quartic in genera $2$ and $3$, respectively, but the asserted uniqueness and global maximality remain conjectural.