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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 normalizeby scaling and adding a constant, to obtain a "mass", which is scale invariantand vanishes at the round sphere. This is an anlaog for closed surfaces of theADM mass from general relativity. We show that if M has positive genus then oneach conformal class, the mass attains a negative minimum. For the minimizingmetric, there is a sharp logarithmic Hardy-Littlewood-Sobolev inequality and aMoser-Trudinger-Onofri type inequality.

A negative mass theorem for surfaces of positive genus