A Spinorial Heat Flow Framework for Geometric Degeneration on $3$-Manifolds

Abstract

We study a spinor-driven formulation of geometric evolution on closed $3$-manifolds, in which the spinor field is treated as the primary dynamical variable and the Riemannian metric is induced conformally by the spinor amplitude. We introduce a spinorial heat flow governed by the squared Dirac operator, \[ \partial_t \psi = - D_{g(\psi)}^{\,2} \psi , \] where the metric $g(\psi)$ depends nonlinearly on the evolving spinor field. As a consequence, the resulting system is quasi-linear and parabolic away from the nodal set $\{\psi=0\}$, while exhibiting degenerate behavior at vanishing spinor amplitude. We show that degeneration of the induced metric corresponds analytically to nodal behavior of the spinor field, rather than to curvature blow-up of the spinor evolution itself. This observation motivates an interpretation of geometric singularities as spinorial nodal transitions, across which the spinor field remains locally bounded in a weak or weighted sense. The induced metric evolution is derived explicitly and shown to be purely conformal, capturing only the trace component of curvature evolution and containing additional gradient terms that are not controlled \emph{a priori}. Accordingly, the proposed flow should not be identified with the Ricci flow, and any analogy with curvature smoothing is understood at a heuristic level. The present work establishes a coherent analytical framework for studying geometric degeneration via spinor dynamics and highlights several open problems in degenerate parabolic theory, including rigorous existence results and the precise role of nodal structures in geometric and topological evolution.