Eigenvalues and Minimal Surfaces

Explore the fascinating world of eigenvalues and minimal surfaces in this illuminating lecture by Mikhail Karpukhin from University College London. Delve into the intriguing connection between Laplace operator eigenvalues and physical phenomena such as heat flow and sound propagation. Trace the history of eigenvalue inequalities from Lord Rayleigh's 19th-century question about the lowest first Dirichlet eigenvalue in planar domains to modern isoperimetric problems involving Riemannian metrics on surfaces. Discover recent advancements in sharp upper bounds for Laplace and Steklov eigenvalues, and their profound link to minimal surface theory. Gain insights into the applications of this connection in both minimal surface theory and isoperimetric eigenvalue problems, culminating in an unexpected relationship between Laplace and Steklov spectra.