A Negative Mass Theorem for Surfaces of Positive Genus

Abstract

Let M be a closed surface. For a metric g on M, denote the Laplace-Beltrami operator by Δ = Δg. We define trace \({\Delta^{-1} = \int_M m(p)dA}\) , where dA is the area element for g and m(p) is the Robin constant at the point \({p \in M}\) , that is the value of the Green function G(p, q) at q = p after the logarithmic singularity has been subtracted off. Since trace Δ−1 can also be obtained by regularization of the spectral zeta function, it is a spectral invariant. Heuristically it represents the sum of squares of the wavelengths of the surface. We define the Δ-mass of (M, g) to equal \({({\rm trace} \Delta_g^{-1} - {\rm trace} \Delta_{S^{2},A}^{-1})/A}\) , where \({\Delta_{S^2,A}}\) is the Laplacian on the round sphere of area A. This is an analog for closed surfaces of the ADM mass from general relativity. We show that if M has positive genus, the minimum of the Δ-mass on each conformal class is negative and attained by a smooth metric. For this minimizing metric, there is a sharp logarithmic Hardy-Littlewood-Sobolev inequality and a Moser-Trudinger-Onofri type inequality.