Abstract:
This talk will be a review of recent results concerning sharp
isoperimetric inequalities for Laplace eigenvalues on surfaces, mainly the
sphere and the real projective plane. The problem of finding the supremum of
Laplace operator eigenvalues on the space of all Riemannian metrics with fixed
area on a surface goes back to a pioneering paper by Hersch in 1970, where this
problem was solved for the first eigenvalue on the sphere. This problem turned out
to be very difficult and till recent years there were results concerning only
several particular cases due to Li, Yau, Nadirashvili, Sire, Petrides et al. In the
recent papers by Karpukhin, Nadirashvili, Penskoi and Polterovich this problem
was completely solved for all eigenvalues on the sphere and the real projective plane.
