An isoperimetric inequality for Laplace eigenvalues on the sphere
Abstract
We show that for any positive integer k, the kth nonzero eigenvalue of the LaplaceBeltrami operator on the twodimensional 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 kth 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2017
 arXiv:
 arXiv:1706.05713
 Bibcode:
 2017arXiv170605713K
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Spectral Theory;
 58J50;
 58E11;
 53C42
 EPrint:
 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