Embedded minimal surfaces in S³ and B³ via equivariant eigenvalue optimization

In 1970, Lawson solved the topological realization problem for minimal surfaces in the sphere, showing that any closed orientable surface can be minimally embedded in S³. The analogous problem for surfaces with boundary was posed by Fraser and Li in 2014, and it has attracted much attention in recent years, stimulating the development of many new constructions for free boundary minimal surfaces. In this paper, we resolve this problem by showing that any compact orientable surface with boundary can be embedded in B³ as a free boundary minimal surface with area below 2. Furthermore, we show that the number of minimal surfaces in S³ of prescribed topology and area below 8, and the number of free boundary minimal surfaces in B³ with prescribed topology and area below 2, grow at least linearly with the genus. This is achieved via a new method for producing minimal surfaces of prescribed topology in low-dimensional balls and spheres, based on the optimization of Laplace and Steklov eigenvalues in the presence of a discrete symmetry group. As a key ingredient, we develop new techniques for proving the existence of maximizing metrics, which can be used to resolve the existence problem in many symmetric situations and provide at least partial existence results for classical eigenvalue optimization problems.