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.
