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

in v2, we simplified the proof of Theorem 3.1We 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