Chern-Ricci flow and t-Gauduchon Ricci-flat condition
Eder M. Correa, Giovane Galindo, Lino Grama
TL;DR
The paper investigates the behavior of the $t$-Gauduchon Ricci-flat condition under the Chern–Ricci flow on non‑Kähler Hermitian manifolds, showing non-preservation for $t<1$ in a class of non‑Kähler Calabi–Yau constructions based on unitary frame bundles over flag manifolds. It develops explicit constructions on principal $T^{2n}$‑bundles over complex flag bases, deriving exact conditions under which the flow preserves or destroys the $t$‑Gauduchon Ricci-flat property, and identifies balanced non‑pluriclosed solutions to the pluriclosed flow. The work also analyzes long‑/short‑time behavior and Gromov–Hausdorff limits of Hermitian flows on these bundles, providing explicit limiting spaces such as torus quotients or circles. The results yield new non‑Kähler Calabi–Yau examples, balance-preserving flows, and concrete collapse phenomena relevant to complex differential geometry and geometric analysis on non‑Kähler manifolds.
Abstract
In this paper, we study the $t$-Gauduchon Ricci-flat condition under the Chern-Ricci flow. In this setting, we provide examples of Chern-Ricci flow on compact non-Kähler Calabi-Yau manifolds which do not preserve the $t$-Gauduchon Ricci-flat condition for $t<1$. The approach presented generalizes some previous constructions on Hopf manifolds. Also, we provide non-trivial new examples of balanced non-pluriclosed solution to the pluriclosed flow on non-Kähler manifolds. Further, we describe the limiting behavior, in the Gromov-Hausdorff sense, of geometric flows of Hermitian metrics (including the Chern-Ricci flow and the pluriclosed flow) on certain principal torus bundles over flag manifolds. In this last setting, we describe explicitly the Gromov-Hausdorff limit of the pluriclosed flow on principal $T^{2}$-bundles over the Fano threefold ${\mathbb{P}}(T_{{\mathbb{P}^{2}}})$.