A negative mass theorem for surfaces of positive genus

Details A negative mass theorem for surfaces of positive genus A negative mass theorem for surfaces of positive genus

2008-10-03

arxiv.org/abs/0810.0724v2

Mathematics

Spectral Theory

8 pages

Scientific paper

10.1007/s00220-008-0722-z

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.

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