Minimally Embedding Compact Surfaces in Round Spheres and Balls (Online)

Abstract

We discuss the topological realization problem for minimally embedding compact surfaces in round spheres and balls. In 1970, using the solution to the Plateau problem, Lawson constructed orientable minimal surfaces of each genus embedded in $S^3$. In recent work with Karpukhin, McGrath and Stern, using equivariant eigenvalue optimization methods and a priori eigenspace dimension bounds, we constructed orientable free boundary minimal surfaces in $B^3$ of any genus and (positive) number of boundary, components. Now we have extended our methods to handle the nonorientable case, constructing embedded minimal surfaces in $S^4$ diffeomorphic to the connect-sum of any number of real projective planes: these all have area (so Willmore bending energy) under $8\pi$ and enjoy other interesting geometric features. The analogous construction for nonorientable free boundary minimal surfaces embedded in $B^4$ looms on the horizon. Open problems and speculation about potential discretizations may be offered if time permits....