Enciso Alberto, Gerner Wadim, Peralta-Salas Daniel
Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, 28049 Madrid, Spain.
Sorbonne Université, Inria, CNRS, Laboratoire Jacques-Louis Lions (LJLL), Paris, France.
Calc Var Partial Differ Equ. 2025;64(5):146. doi: 10.1007/s00526-025-02995-7. Epub 2025 May 5.
In this article we analyze the spectral properties of the curl operator on closed Riemannian 3-manifolds. Specifically, we study metrics that are optimal in the sense that they minimize the first curl eigenvalue among any other metric of the same volume in the same conformal class. We establish a connection between optimal metrics and the existence of minimizers for the -norm in a fixed helicity class, which is exploited to obtain necessary and sufficient conditions for a metric to be locally optimal. As a consequence, our main result is that we prove that and endowed with the round metric are -local minimizers for the first curl eigenvalue (in its conformal and volume class). The connection between the curl operator and the Hodge Laplacian allows us to infer that the canonical metrics of and are locally optimal for the first eigenvalue of the Hodge Laplacian on coexact 1-forms. This is in strong contrast to what happens in four dimensions.
在本文中，我们分析了闭黎曼三维流形上旋度算子的谱性质。具体而言，我们研究这样的度量：在同一共形类中具有相同体积的任何其他度量中，它们能使第一旋度特征值最小化，从这个意义上来说是最优的。我们建立了最优度量与固定螺旋度类中 - 范数极小值的存在性之间的联系，利用这一联系来获得度量为局部最优的充分必要条件。因此，我们的主要结果是证明赋予标准度量的 和 是第一旋度特征值（在其共形类和体积类中）的 - 局部极小值。旋度算子与霍奇拉普拉斯算子之间的联系使我们能够推断出，对于余恰当1 - 形式上的霍奇拉普拉斯算子的第一特征值， 和 的标准度量是局部最优的。这与四维情形形成了强烈对比。
