A Chern–Calabi Flow on Hermitian Manifolds

Shen, Xi Sisi

doi:10.1007/s12220-021-00845-4

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2023

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“…If the initial data are Hermitian but not necessarily Kähler and

\operatorname{Ric}\omega

is interpreted as the Ricci curvature of the Chern connection, then Equation (1.1) defines the Chern–Ricci flow introduced by Gill [21]. There is a plethora of the literature on the study of non‐Kähler flows [1, 4, 7, 8, 10, 14, 15, 17–19, 27, 31, 41, 44, 45, 53, 56–58, 60, 61].…”

Section: Introductionmentioning

confidence: 99%

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

Liang,

Shen,

Smith

2023

Bulletin of London Math Soc

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We prove local Calabi and higher order estimates for solutions to the continuity equation introduced by La Nave–Tian and extended to Hermitian metrics by Sherman–Weinkove. We apply the estimates to show that on a compact complex manifold, the Chern scalar curvature of a solution must blow up at a finite‐time singularity. Additionally, starting from certain classes of initial data on Oeljeklaus–Toma manifolds, we prove Gromov–Hausdorff and smooth convergence of the metric to a particular nonnegative (1,1)‐form as .

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“…If the initial data are Hermitian but not necessarily Kähler and