关于大k极小谱k-分拆和Pleijel定理的综述
A review on large k minimal spectral k-partitions and Pleijel's Theorem
摘要：
在这篇综述中，我们回顾了二维情形下的最小谱$k$ -划分的性质，并重新审视了它们与Pleijel定理的联系。本文结合Bourgain和S. Steinerberger关于Pleijel定理这使我们也讨论一些猜想我。Polterovich，在与正方形瓷砖。我们还建立了Aharonov-Bohm哈密顿算子的Pleijel定理，并通过极小分拆的磁特征，从该定理导出极小分拆的临界点个数的下界.
英文摘要：
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.
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