An Overdetermined Eigenvalue Problem and the Critical Catenoid Conjecture

Explore a seminar on spectral geometry focusing on an overdetermined eigenvalue problem and its connection to the Critical Catenoid conjecture. Delve into José Espinar's research from the University of Granada, presented as part of the "Spectral Geometry in the clouds" series. Examine the eigenvalue problem ∆S 2 ξ + 2ξ = 0 in Ω with ξ = 0 along ∂Ω, where Ω is the complement of disjoint, finite unions of smooth, bounded, simply connected regions on the two-sphere S 2. Investigate the conditions under which positive solutions can be classified as rotationally symmetric, including locally constant k∇ξk along ∂Ω and infinitely many maximum points for ξ. Discover how this analysis leads to a characterization of the critical catenoid as the unique embedded free boundary minimal annulus in the unit ball with a support function possessing infinitely many critical points.