We show that for any positive integer k, the k-th nonzero eigenvalue of the Laplace-Beltrami operator on the two-dimensional sphere endowed with a Riemannian metric of unit area, is maximized in the limit by a sequence of metrics converging to a union of k touching identical round spheres. This proves a conjecture posed by the second author in 2002 and yields a sharp isoperimetric inequality for all nonzero eigenvalues of the Laplacian on a sphere. Earlier, the result was known only for k=1 (J. Hersch, 1970), k=2 (N. Nadirashvili, 2002; R. Petrides, 2014) and k=3 (N. Nadirashvili and Y. Sire, 2017). In particular, we argue that for any k>=2, the supremum of the k-th nonzero eigenvalue on a sphere of unit area is not attained in the class of Riemannian metrics which are smooth outsitde a finite set of conical singularities. The proof uses certain properties of harmonic maps between spheres, the key new ingredient being a bound on the harmonic degree of a harmonic map into a sphere obtained by N. Ejiri.

中文翻译：

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球面上拉普拉斯特征值的等周不等式

我们证明，对于任何正整数 k，具有单位面积黎曼度量的二维球体上的 Laplace-Beltrami 算子的第 k 个非零特征值通过收敛到并集的一系列度量在极限内最大化k 接触相同的圆形球体。这证明了第二作者在 2002 年提出的猜想，并为球体上拉普拉斯算子的所有非零特征值产生了一个尖锐的等周不等式。此前，只有 k=1 (J. Hersch, 1970)、k=2 (N. Nadirashvili, 2002; R. Petrides, 2014) 和 k=3 (N. Nadirashvili and Y. Sire, 2017) 才知道结果. 特别是，我们认为，对于任何 k>=2，单位面积球体上的第 k 个非零特征值的上确界在黎曼度量的类中没有达到，这些度量在有限的圆锥奇点集之外是平滑的。