A Negative Mass Theorem for the 2-Torus

Abstract

Let M be a closed surface. For a metric g on M, denote the area element by dA and the Laplace-Beltrami operator by Δ = Δ g . We define the Robin mass m(p) at the point $${{p \in\, M}}$$ to be the value of the Green function G(p, q) at q = p after the logarithmic singularity has been subtracted off, and we define trace $${{\Delta^{-1}=\int_M m(p)\, dA}}$$ . This regularized trace can also be obtained by regularization of the spectral zeta function and is hence a spectral invariant which heuristically measures the total wavelength 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 scale invariant quantity is a non-trivial analog for closed surfaces of the ADM mass for higher dimensional asymptotically flat manifolds.In this paper we show that in each conformal class $${{\mathcal C}}$$ for the 2-torus, there exists a metric with negative Δ-mass. From this it follows that the minimum of the Δ-mass on $${{\mathcal C}}$$ is negative and attained by some metric $${{g\in\mathcal C}}$$ . For this minimizing metric g, one gets a sharp logarithmic Hardy-Littlewood-Sobolev inequality and an Onofri-type inequality.We remark that if the flat metric in $${{\mathcal C}}$$ is sufficiently long and thin then the minimizing metric g is non-flat. The proof of our result depends on analyzing the ordinary differential equation $${{\phi''=1-e^\phi}}$$ which is equivalent to h′′ = 1 − 1/h. The solutions are periodic and we need to establish quite delicate, asymptotically sharp inequalities relating the period to the maximum value.