Critical metrics for Laplace eigenvalues and their connection to minimal and harmonic maps

In “The Theory of Sound” (1877–1878), Lord Rayleigh posed the question: Which shape of a drum membrane yields the lowest fundamental frequency among all membranes of fixed area? Using physical intuition, he conjectured the answer to be the disc — an assertion later rigorously proven by Faber and Krahn in 1921. A modern analogue in Riemannian geometry asks: given a compact surface without boundary and a positive integer $k$, what is the supremum of the $k$th eigenvalue of the Laplace–Beltrami operator over all Riemannian metrics of fixed area? This problem is both challenging and rich in structure, with deep connections to classical areas such as differential and algebraic geometry, geometric analysis, partial differential equations, and topology. In particular, the interplay between critical metrics for eigenvalues and minimal or harmonic maps has emerged as a powerful and insightful framework.