**Maximizers beyond the hemisphere for the second Neumann eigenvalue **

Abstract: Among membranes of given area vibrating freely on a sphere, which shape achieves the highest fundamental tone? Equivalently, on which insulated subdomain of given size in a sphere will the temperature equilibrate most rapidly? Motivated by symmetry, one conjectures the answer is a spherical cap.

We prove the conjecture for all but the largest domains by showing that the first positive Neumann eigenvalue mu_2 of the spherical Laplacian is maximal for a spherical cap among simply connected domains covering up to 94% of the 2-sphere. The previous best result, by Bandle, was 50%, which means her maximizing caps lay in a hemisphere.

Further questions includeâ€¦what if the domain has holes? and is the next positive eigenvalue mu_3 maximal for the union of two disjoint caps, analogous to the two-disk result for planar domains?

(Joint work with Jeffrey Langford, Bucknell University)