Karpukhin, Mikhail; Nadirashvili, Nikolai; Penskoi, Alexei V; Polterovich, Iosif; (2021) An isoperimetric inequality for Laplace eigenvalues on the sphere. Journal of Differential Geometry , 118 (2) pp. 313-333. 10.4310/jdg/1622743142.

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Abstract

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 outside 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.

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