Spinors and mass on weighted manifolds
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.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 July 2022
 DOI:
 10.1007/s0022002204420y
 arXiv:
 arXiv:2201.04475
 Bibcode:
 2022CMaPh.tmp..157B
 Keywords:

 Mathematics  Differential Geometry;
 Mathematical Physics;
 Mathematics  Analysis of PDEs
 EPrint:
 19 pages