%0 Journal Article
%T A negative mass theorem for surfaces of positive genus
%A Kate Okikiolu
%J Mathematics
%D 2008
%I arXiv
%R 10.1007/s00220-008-0722-z
%X We define the "sum of squares of the wavelengths" of a Riemannian surface (M,g) to be the regularized trace of the inverse of the Laplacian. We normalize by scaling and adding a constant, to obtain a "mass", which is scale invariant and vanishes at the round sphere. This is an anlaog for closed surfaces of the ADM mass from general relativity. We show that if M has positive genus then on each conformal class, the mass attains a negative minimum. For the minimizing metric, there is a sharp logarithmic Hardy-Littlewood-Sobolev inequality and a Moser-Trudinger-Onofri type inequality.
%U http://arxiv.org/abs/0810.0724v2