On the stability of the anomaly flow (Q2079509)

Let \(M\) be a compact complex manifold of complex dimension greater than or equal to three. Let \(\Omega\) be a fixed holomorphic volume form. The anomaly flow with zero slope parameter is the flow of hermitian metrics \(\omega(t)\) given by \[ \partial_t (|\Omega|_\omega \, \omega^{n-1}) = i \partial \bar{\partial} \omega^{n-2}. \] The authors prove the stability of this flow about Calabi-Yau metrics. That is, starting the flow with an initial metric which is nearby a Kähler Calabi-Yau metric gives long-time existence and convergence of the flow. The proof involves the fact that the linearization of the flow at a Kähler Calabi-Yau metric is the Laplacian acting on \((n-1,n-1)\) forms. The authors then state and prove a general stability theorem for geometric flows with an integrability condition. This general argument uses the Hamilton-Nash-Moser theorem.