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.
- Pub Date:
- June 2017
- Mathematics - Differential Geometry;
- Mathematics - Spectral Theory;
- The expository part of the paper draws heavily on the earlier preprint arXiv:1608.07334 by the second and the third authors. v2: Major revision, the statement of the main result is revised, the proof is simplified. v3: Minor revision. The bibliography is updated, a misprint in Theorem 2.3 is corrected. v4: Minor revision. The bibliography is updated. LaTeX, 19 pages