Relativistic Hamilton–Perelman geometric flow theory for metric-affine (nonmetric) distortions

Establish rigorous formulations and proofs of relativistic variants of Hamilton–Perelman-type geometric flow results (including appropriate F- and W-functionals and associated evolution equations) for metric-affine geometries with nonmetricity and torsion, where connections are distorted away from Levi-Civita, thereby extending the Riemannian theory to nonmetric modified gravity settings.

Background

The authors generalize Perelman's F- and W-functionals to nonmetric Einstein–Dirac–Maxwell systems under metric-affine distortions. While they define functional analogues suitable for their canonical nonholonomic framework, they explicitly state that the broader mathematical program of formulating and proving relativistic variants of Hamilton–Perelman conjectures and related theorems for general metric-affine (nonmetric) connections remains unclear.

This reflects a gap between the well-developed Riemannian Ricci flow theory and potential extensions to metric-affine settings with nonmetricity and torsion, where a rigorous foundation analogous to the Poincaré–Thurston program is not yet established.