A renormalized Perelman-functional and a lower bound for the ADM-mass

In the first part of this short article, we define a renormalized FF-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 λAFλAF 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.

► We define a renormalized FF-functional for non-compact manifolds. ► The FF-functional motivates a stability inequality for Ricci-flat spaces. ► We define a geometric invariant lambda for asymptotically flat manifolds. ► The lambda invariant gives a quantitative lower bound for the ADM-mass. ► We announce the discovery of a mass-decreasing flow in dimension three.