Isoperimetric inequality for the third eigenvalue of the Laplace-Beltrami operator on ${\mathbb{S}}^2$

Abstract

We prove an Hersch's type isoperimetric inequality for the third positive eigenvalue on ${\mathbb{S}}^2$. Our method builds on the theory we developped to construct extremal metrics on Riemannian surfaces in conformal classes for any eigenvalue.

in v2, we simplified the proof of Theorem 3.1

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