With A. Deruelle, we define a Perelman like functional for ALE metrics which lets us study the (in)stability of Ricci-flat ALE metrics. With J. Baldauf, we extend some classical objects and formulas from the study of scalar curvature, spin geometry and general relativity to manifolds with densities. We surprisingly find that the extension of ADM mass is the opposite of the above functional introduced with A. Deruelle. Through a weighted Witten’s formula, this functional also equals a weighted spinorial Dirichlet energy on spin manifolds. Ricci flow is the gradient flow of all of these quantities.