  In the first part of this short article, we define a renormalized
F-functional for perturbations of non-compact steady Ricci solitons. This
functional motivates a stability inequality which plays an important role in
questions concerning the regularity of Ricci-flat spaces and the non-uniqueness
of the Ricci flow with conical initial data. In the second part, we define a
geometric invariant lambda_AF for asymptotically flat manifolds with
nonnegative scalar curvature. This invariant gives a quantitative lower bound
for the ADM-mass from general relativity, motivates a Ricci flow proof of the
rigidity statement in the positive mass theorem, and eventually leads to the
discovery of a mass decreasing flow in dimension three.
