Extremal Laplace Eigenvalues on Riemann Surfaces: A Comprehensive Variational Approach over Moduli Space

Abstract

We investigate the extremal values of Laplace-Beltrami eigenvalues on compact, orientable Riemann surfaces of genus , under normalization of total area and within fixed or varying conformal classes. By combining variational methods on the moduli space  with analytic perturbation theory, asymptotic analysis, and the theory of harmonic maps, we establish new sharp upper bounds for the first non-zero eigenvalue  and characterize critical metrics as induced by minimal immersions into spheres. We prove the existence and uniqueness of smooth extremal metrics in each conformal class through second variation analysis, and demonstrate the continuity of the eigenvalue functional over the Deligne-Mumford compactification of , and provide comprehensive asymptotic behavior near boundary degenerations via Green's function expansions. We further generalize our framework to higher eigenvalues  with complete minmax characterizations, nodal set analysis, and harmonic map extensions. Extensive numerical evidence for genus- 2 extremal metrics is provided via finite element simulations with rigorous error analysis, mesh refinement studies, and multiplicity tracking on special surfaces.