Title: An isoperimetric inequality for Laplace eigenvalues on the sphere,
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
Riemannin 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.