Harmonic spinors in the Ricci flow

TL;DR

This paper links Ricci flow on closed manifolds with harmonic spinors, expressing Perelman's entropy via spinor energies and establishing new inequalities, revealing singularities in exotic K3 surfaces.

Contribution

It introduces a weighted monopole equation framework and connects Ricci flow to spinor energies, providing new insights into four-dimensional geometry.

Findings

Perelman's entropy expressed via harmonic spinor energy

A sharp parabolic Hitchin-Thorpe inequality proven for spin 4-manifolds

Normalized Ricci flow on exotic K3 surfaces must develop singularities

Abstract

This paper studies the Ricci flow on closed manifolds admitting harmonic spinors. It is shown that Perelman's Ricci flow entropy can be expressed in terms of the energy of harmonic spinors in all dimensions, and in four dimensions, in terms of the energy of Seiberg-Witten monopoles. Consequently, Ricci flow is the gradient flow of these energies. The proof relies on a weighted version of the monopole equations, introduced here. Further, a sharp parabolic Hitchin-Thorpe inequality for simply-connected, spin 4-manifolds is proven. From this, it follows that the normalized Ricci flow on any exotic K3 surface must become singular.