The study of sharp upper bounds for Laplacian eigenvalues under area constraints is a classical topic in spectral geometry. A key source of interest lies in the remarkable fact that metrics achieving equality in such bounds correspond to metrics induced by minimal surfaces in spheres. Even more strikingly, analogous connections between eigenvalues and natural geometric structures continue to emerge across diverse settings.
In this talk, I will highlight notable examples of these correspondences and discuss their implications for both spectral estimates and geometric analysis.
