Existence and Uniqueness for Free Boundary Minimal Surfaces

Abstract: Let B^3 be the unit ball in R^3 and consider the family of surfaces contained in B^3 whose boundaries lie on the unit sphere S^2.  The critical points of the area functional amongst this class are called Free Boundary Minimal Surfaces.  The latter surfaces are physically realized by soap films in equilibrium and have interesting geometric and analytic interpretations.  In the 1980s, it was proved that flat equatorial disks are the only free boundary minimal surfaces with the topology of a disk.  It is conjectured that a surface called the critical catenoid is the unique (up to ambient rotations) free boundary minimal annulus.  I will discuss some recent progress towards resolving this conjecture.