  This paper introduces a functional generalizing Perelman's weighted
Hilbert-Einstein action and the Dirichlet energy for spinors. It is
well-defined on a wide class of non-compact manifolds; on asymptotically
Euclidean manifolds, the functional is shown to admit a unique critical point,
which is necessarily of min-max type, and Ricci flow is its gradient flow. The
proof is based on variational formulas for weighted spinorial functionals,
valid on all spin manifolds with boundary.
