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An isoperimetric inequality for Laplace eigenvalues on the sphere

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 Added by Alexei V. Penskoi
 Publication date 2017
  fields
and research's language is English




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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.



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The paper is concerned with the maximization of Laplace eigenvalues on surfaces of given volume with a Riemannian metric in a fixed conformal class. A significant progress on this problem has been recently achieved by Nadirashvili-Sire and Petrides using related, though different methods. In particular, it was shown that for a given $k$, the maximum of the $k$-th Laplace eigenvalue in a conformal class on a surface is either attained on a metric which is smooth except possibly at a finite number of conical singularities, or it is attained in the limit while a bubble tree is formed on a surface. Geometrically, the bubble tree appearing in this setting can be viewed as a union of touching identical round spheres. We present another proof of this statement, developing the approach proposed by the second author and Y. Sire. As a side result, we provide explicit upper bounds on the topological spectrum of surfaces.
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