Geometry and Topology Seminar

It has been conjectured that the first nontrivial eigenvalue of the Dirichlet-to-Neumann map on an embedded free boundary minimal surface in the unit 3-ball is one. I will discuss recent work with R. Kusner which provides sufficient criteria for the first eigenvalue on such a surface to be equal to one, and moreover that the corresponding eigenspace is spanned by the coordinate functions. A consequence of this work is that an embedded antipodally invariant free boundary minimal annulus in the unit ball is congruent to the critical catenoid.