Geometry Seminar: M. Karpukhin (UCL)

GeoTop Geometry Seminar (Geometric Analysis)

Speaker: Mikhail Karpukhin (University College London)

Title: New embedded minimal surfaces in 3-sphere and 3-ball via eigenvalue optimisation.

Abstract: The study of optimal upper bounds for Laplace eigenvalues on closed surfaces under area constraint is a classical problem of spectral geometry. It is particularly interesting due to the fact that optimal metrics (if exist) correspond to branched minimal surfaces in n-dimensional sphere. In general, determining whether such metrics exist, whether the corresponding maps are embeddings, and determining the dimension of the sphere are challenging problems, where very few results are known. In the present talk we will discuss how one can use group action to resolve these issues and, as a result, construct many new examples of embedded minimal surfaces in the 3-sphere. The same considerations can be applied to the Steklov eigenvalue problem. As a consequence, we completely resolve the realisation problem for free boundary minimal surfaces in the unit 3-ball: we show that any compact orientable surface with boundary can be embedded in the 3-ball as a free boundary minimal surface. Based on a joint work with R. Kusner, P. McGrath and D. Stern.