In 1996, Nadirashvili discovered a beautiful connection between minimal surfaces in round spheres and an optimization problem for Laplace eigenvalues on a surface. This gave a surprising analytic perspective on minimal surfaces and opened the way for many important results.

Here, we will focus on the recent paper from 2024 by Karpukhin–Kusner–McGrath–Stern, who use equivariant eigenvalue optimization to construct many new examples of minimal surfaces in the three-dimensional unit sphere $\mathbb S^3$. Using similar methods, they also find many new free boundary minimal surfaces in the three-dimensional unit ball $\mathbb B^3$, in particular obtaining examples for every topological type. This was a central open problem in the field, posed by Fraser–Li in 2014, whose analogue in $\mathbb S^3$ was solved by Lawson in 1970. Note that free boundary minimal surfaces in round balls enjoy a connection with another eigenvalue problem, namely the Steklov problem, by a result of Fraser–Schoen.

In the talk, we will give an overview of the results. We will present the ingredients in the proof of Karpukhin–Kusner–McGrath–Stern (including previous results by Petrides from 2014 and Karpukhin–Stern from 2020) and we will focus on the novel techniques of the paper. These have already spurred important advances in the study of Laplace eigenvalue optimization by Petrides (2024) and Karpukhin–Petrides–Stern (2025).

[After Karpukhin, Kusner, McGrath, and Stern]