I'll discuss the papers "Berger's isoperimetric problem and minimal
immersions of surfaces" by Nadirashvili, "Extremal metric for the first
eigenvalue on a Klein Bottle" by Jakobson, Nadirashvili and Polterovich
and "How large can the first eigenvalue be on a surface of genus two?" by
Jakobson, Levitin, Nadirashvili, Nigam and Polterovich. The authors find
among metrics of fixed area one that maximizes the fundamental frequency
of the torus and Klein Bottle.