The spinorial energy for asymptotically Euclidean Ricci flow

TL;DR

This paper introduces a new spinorial energy functional for asymptotically Euclidean manifolds, showing it admits a unique critical point and that Ricci flow acts as its gradient flow, extending geometric analysis tools.

Contribution

It generalizes Perelman's functional to spinors on non-compact manifolds and establishes its critical points and relation to Ricci flow.

Findings

Unique critical point of the functional on asymptotically Euclidean manifolds

Ricci flow is the gradient flow of the introduced functional

Variational formulas valid on all spin manifolds with boundary

Abstract

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.