The pseudo-Calabi flow

Abstract

We define the pseudo-Calabi flow as qj qt ¼ Àf ðjÞ, h j f ðjÞ ¼ SðjÞ À S.Then we prove the well-posedness of this flow including the short time existence, the regularity of the solution and the continuous dependence on the initial data. Next, we point out that the L y bound on Ricci curvature is an obstruction to the extension of the pseudoCalabi flow. Finally, we show that if there is a constant scalar curvature Kähler metric o in its Kähler class, then for any initial potential in a small C 2; a neighborhood of this metric (defined in terms of the C 2; a norm on the Kähler potential), the pseudo-Calabi flow must converge exponentially fast to a constant scalar curvature Kähler metric near o within the same Kähler class.

Propagation of chaos, Wasserstein gradient flows and toric Kähler–Einstein metrics

“…In particular, as in [2] convexity (or more generally λ−convexity) plays a prominent role. Our limiting evolution equation will appear as the gradient flow on P 2 (R n ) of a certain free energy type functional F. Interestingly, as observed in [7] the functional F may be identified with Mabuchi's K-energy functional on the space of Kähler metrics, which plays a key role in Kähler geometry and whose gradient flow with respect to different metrics (the Mabuchi-Donaldson-Semmes metric and Calabi's gradient metric) are the renowned Calabi flow and Kähler-Ricci flow, respectively [28]. The regularity and large time properties of the evolution equation appearing here will be studied elsewhere [13,12].…”

Geometric flows and Kähler reduction

“…for some constant b ∈ R and some smooth function u on M . Substituting in the first equation of (3) then proves (2). It remains to show that any Kähler form ω on P making Hamiltonian the standard circle action with moment map µ satisfies (i) and (ii) for some triple (σ, φ, c).…”

From the Kähler-Ricci flow to moving free boundaries and shocks

“…In Section 7.4 below we will give a new interpretation of the metric 7.2, motivated by probabilistic considerations. The relation between the KRF and gradient flows wrt the Dirichlet metric first appeared in [29] (in the non-twisted setting).…”