Extremal Domains for the First Eigenvalue of the Laplacian

The Faber-Krahn inequality asserts that the round balls uniquely minimize the first eigenvalue of the Laplacian among domains in $\R^n$ with fixed volume.  In 1970, Serrin proved more generally that balls are the only critical points for the functional assigning a domain in $\R^n$ to its first eigenvalue, among volume-preserving variations.  Much less is known about the analogous problem for domains in the round $n$-sphere, and for $n=2$, Souam conjectured (2005) the only critical domains are rotationally symmetric disks and annuli.  I will discuss the construction of counterexamples (joint work with C. Hines and J. Kolesar) to this conjecture.