On the pluriclosed flow on Oeljeklaus–Toma manifolds

Abstract

We investigate the pluriclosed flow on Oeljeklaus-Toma manifolds. We parametrize leftinvariant pluriclosed metrics on Oeljeklaus-Toma manifolds and we classify the ones which lift to an algebraic soliton of the pluriclosed flow on the universal covering. We further show that the pluriclosed flow starting from a left-invariant pluriclosed metric has a long-time solution ωt which once normalized collapses to a torus in the Gromov-Hausdorff sense. Moreover the lift of 1 1+t ωt to the universal covering of the manifold converges in the Cheeger-Gromov sense to (H s × C s , ω∞) where ω∞ is an algebraic soliton.

The continuity equation for Hermitian metrics: Calabi estimates, Chern scalar curvature, and Oeljeklaus–Toma manifolds

“…It was shown in [36] that OT manifolds do not admit any balanced metrics but always have locally conformally balanced metrics. See [2,3,20,25,34,35,37,[62][63][64] for recent progress and open problems on OT manifolds. Using the local Calabi estimates of Theorem 1, alongside an adaptation of a rescaling argument used by Angella-Tosatti [4] in the setting of Inoue surfaces, we can prove higher order estimates and smooth convergence of solutions to the continuity equation starting from a certain class of initial data.…”