  We prove a sharp isoperimetric inequality for the second nonzero eigenvalue
of the Laplacian on $S^m$. For $S^{2}$, the second nonzero eigenvalue becomes
maximal as the surface degenerates to two disjoint spheres, by a result of
Nadirashvili for which Petrides later gave another proof. For higher
dimensional spheres, the analogous upper bound was conjectured by Girouard,
Nadirashvili and Polterovich. Our method to confirm the conjecture builds on
Petrides' work and recent developments on the hyperbolic center of mass and
provides also a simpler proof for $S^2$.
