Leafwise flat forms on Inoue-Bombieri surfaces

Abstract

We prove that every Gauduchon metric on an Inoue-Bombieri surface admits a strongly leafwise flat form in its ∂∂-class. Using this result, we deduce uniform convergence of the normalized Chern-Ricci flow starting at any Gauduchon metric on all Inoue-Bombieri surfaces. We also show that the convergence is smooth with bounded curvature for initial metrics in the ∂∂-class of the Tricerri/Vaisman metric.

Almost complex Hodge theory

“…This method was used to study geometric problems on nilmanifolds in e.g. [3,9,24] and recently [25,1]. The new ingredients in [16,17] are the roles played by Stokes phenomenon and Gauss circle problem.…”

“…We take J a,b and the standard orthonormal metrics for example. We write (1, 1)-form as s = f (1,1) φ 1 ∧ φ1 + f (1,2) φ 1 ∧ φ2 + f (2,1) φ 2 ∧ φ1 + f (2,2) φ 2 ∧ φ2 .…”

“…When n = 0, we will get three dimensions of solutions f (1,1) = C 0 , f (1,2) = C 1 , f (2,1) = −C 1 , f (2,2) = C 2 .…”

On the pluriclosed flow on Oeljeklaus–Toma manifolds

“…In [2,7,29,32] the Chern-Ricci flow [10,28] on Oeljeklaus-Toma manifolds M of type (r, 1) is studied. Accordingly to the results in [2,7,29,32], under some assumptions on the initial Hermitian metric, the flow has a long-time solution ω t such that (M, ωt 1+t ) converges in the Gromov-Hausdorff sense to an r-dimensional torus T r as t → ∞.…”

“…In [2,7,28,32], the Chern-Ricci flow [10,29] on Oeljeklaus-Toma manifolds M of type (r, 1) is studied. According to the results in [2,7,28,32], under some assumptions on the initial Hermitian metric, the flow has a long-time solution ω t such that (M, ωt 1+t ) converges in the Gromov-Hausdorff sense to an r-dimensional torus T r as t → ∞. The result can be adapted to Oeljeklaus-Toma manifolds of arbitrary type by assuming 40 E. Fusi and L. Vezzoni the initial metric to be left-invariant with respect to the structure of solvmanifold.…”