Geometry and Topology of Spectral Minimal Partitions

A minimal partition is a decomposition of a manifold into disjoint sets that minimizes a spectral energy functional. In the bipartite case minimal partitions are closely related to eigenfunctions of the Laplacian, but in the non-bipartite case they are difficult to classify, even for simple domains like the square or the circle.

I will present new results that imply, among other things, that a partition that minimizes energy locally is in fact a global minimum (in the bipartite case) and a minimum within a certain topological class of partitions in the non-bipartite case. Understanding this topological class of partitions requires the introduction of a modified Laplacian operator, or an equivalent Aharonov-Bohm Hamiltonian with singular magnetic potential. I will also explain how to construct energy-decreasing deformations of partitions that are critical but not minimal, giving insight into the geometric structure of the true minimum. This talk is based on joint work with Gregory Berkolaiko, Yaiza Canzani, Peter Kuchment and Jeremy Marzuola.