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 $k \geqslant 2$, 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.
M. Karpukhin was partially supported by the Tomlinson Fellowship and the Schulich Fellowship.
A. V. Penskoi was partially supported by the Simons–IUM Fellowship and by the Young Russian Mathematics award.
I. Polterovich was partially supported by NSERC, FRQNT, and the Canada Research Chairs program.
"An isoperimetric inequality for Laplace eigenvalues on the sphere." J. Differential Geom. 118 (2) 313 - 333, June 2021. https://doi.org/10.4310/jdg/1622743142