Seminar talk, 14 March 2018

From Geometry of Differential Equations
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Speaker: Alexei Penskoi

Title: An isoperimetric inequality for Laplace-Beltrami eigenvalues on the sphere

Abstract:
In the talk we prove isoperimetric inequality for all nonzero eigenvalues of the Laplace-Beltrami operator on a sphere: for any positive integer [math]\displaystyle{ k }[/math], the eigenvalue [math]\displaystyle{ \lambda_k }[/math] 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 [math]\displaystyle{ k }[/math] touching identical round spheres with standard metric.

This proves a conjecture posed by Nadirashvili in 2002 and yields a sharp isoperimetric inequality for all nonzero eigenvalues of the Laplace-Beltrami operator on a sphere. Earlier, the result was known only for [math]\displaystyle{ k=1 }[/math] (Hersch, 1970), [math]\displaystyle{ k=2 }[/math] (Nadirashvili, 2002; Petrides, 2014) and [math]\displaystyle{ k=3 }[/math] (Nadirashvili and Sire, 2017). In particular, this means that for any [math]\displaystyle{ k\ge 2 }[/math], the supremum of the [math]\displaystyle{ k }[/math]-th nonzero eigenvalue on a sphere of unit area is not attained in the class of Riemannin metrics which are smooth outside a finite set of conical singularities. The proof uses certain properties of harmonic maps between spheres.

Note that for the first time the geometric optimization problem for eigenvalues is solved for all eigenvalues: previously this problem was solved only for some eigenvalues.

Joint work with Mikhail Karpukhin, Nikolai Nadirashvili, and Iosif Polterovich.

References:
arXiv:1706.05713