Seminar

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Deep connections between extremal eigenvalue problems and minimal surfaces in round spheres or balls have emerged over the past few decades, stemming from work of Nadirashvili and Fraser-Schoen.  We use these connections to determine geometric properties of these surfaces, such as a uniqueness theorem for embedded free boundary minimal annuli with antipodal symmetry, by showing the first eigenspace for many of these surfaces coincides with the span of the ambient coordinate functions (a generalization of the Yau and Fraser-Li conjectures).   We also develop an equivariant version of eigenvalue optimization with a sharp a priorieigenspace dimension bound, letting us construct free boundary minimal surfaces of every topological type embedded in the round 3-ball, and many more closed minimal surfaces embedded in the round 3-sphere. 

[based on joint projects with Misha Karpukhin, Peter McGrath and Daniel Stern]

On Minimal Surfaces in Round Spheres and Balls