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 $k \geqslant 2$, 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.