A review on large k minimal spectral k-partitions and Pleijel's Theorem

TL;DR

This survey explores large k minimal spectral partitions in 2D, their relation to Pleijel's Theorem, recent conjectures, and extends Pleijel's Theorem to Aharonov-Bohm Hamiltonians, highlighting new bounds and connections.

Contribution

It reviews properties of minimal spectral k-partitions, discusses the large k problem and hexagonal conjecture, and establishes a new Pleijel Theorem for Aharonov-Bohm Hamiltonians.

Findings

Established a Pleijel Theorem for Aharonov-Bohm Hamiltonians.

Connected minimal spectral partitions with Pleijel's Theorem and recent conjectures.

Derived lower bounds for the number of critical points in minimal partitions.

Abstract

In this survey, we review the properties of minimal spectral $k$-partitions in the two-dimensional case and revisit their connections with Pleijel's Theorem. We focus on the large $k$ problem (and the hexagonal conjecture) in connection with two recent preprints by J. Bourgain and S. Steinerberger on the Pleijel Theorem. This leads us also to discuss some conjecture by I. Polterovich, in relation with square tilings. We also establish a Pleijel Theorem for Aharonov-Bohm Hamiltonians and deduce from it, via the magnetic characterization of the minimal partitions, some lower bound for the number of critical points of a minimal partition.