  For a closed surface M with metric g, the Robin mass m(p) at the point p is
the value of the Green function G(p,q) at p=q after the logarithmic singularity
has been removed. The Laplacian-mass is the average value of the Robin mass,
minus the value of the Robin mass for the round sphere of the same area. The
Laplacian-mass is a spectral invariant which is a natural analog of the ADM
mass for asymptotically flat manifolds. We show that if M is a torus, then the
minimum value of the Laplacian-mass on the conformal class of g is negative. It
is attained by a (smooth) metric for which one gets a sharp logarithmic
Hardy-Littlewood-Sobolev inequality and Onofri-type inequality. If the flat
metric in the conformal class is sufficiently long and thin, then the minimizer
for the Laplacian-mass is non-flat.
