A negative mass theorem for surfaces of positive genus

TL;DR

This paper introduces a conformal invariant called 'mass' for Riemannian surfaces, showing it is negative for surfaces with positive genus and establishing related inequalities for the minimizing metrics.

Contribution

It defines a new scale-invariant mass for surfaces, proves it attains negative minima on positive genus surfaces, and derives sharp inequalities for the minimizing metrics.

Findings

Mass is negative for surfaces with positive genus.

Existence of sharp inequalities for the minimizing metrics.

Mass vanishes at the round sphere.

Abstract

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.