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.

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