Spinors and mass on weighted manifolds

TL;DR

This paper extends spin geometry to weighted manifolds, exploring spectral properties of weighted Dirac operators, defining a weighted ADM mass, and linking Ricci flow to the monotonicity of a weighted spinorial energy, with applications to geometric analysis.

Contribution

It introduces a generalized weighted spin geometry framework, defines a weighted ADM mass, and connects Ricci flow with weighted spinorial energies, providing new tools for geometric analysis on weighted manifolds.

Findings

Weighted Dirac operator spectral properties analyzed.

A weighted ADM mass satisfying a positive mass theorem established.

Ricci flow shown to be the gradient flow of weighted ADM mass.

Abstract

This paper generalizes classical spin geometry to the setting of weighted manifolds (manifolds with density) and provides applications to the Ricci flow. Spectral properties of the naturally associated weighted Dirac operator, introduced by Perelman, and its relationship with the weighted scalar curvature are investigated. Further, a generalization of the ADM mass for weighted asymptotically Euclidean (AE) manifolds is defined; on manifolds with nonnegative weighted scalar curvature, it satisfies a weighted Witten formula and thereby a positive weighted mass theorem. Finally, on such manifolds, Ricci flow is the gradient flow of said weighted ADM mass, for a natural choice of weight function. This yields a monotonicity formula for the weighted spinorial Dirichlet energy of a weighted Witten spinor along Ricci flow.