On the evolution of a Hermitian metric by its Chern-Ricci form

By Valentino Tosatti, B. Weinkove. 2011

150 citations recorded by the source corpus.

We consider the evolution of a Hermitian metric on a compact complex manifold by its Chern-Ricci form. This is an evolution equation first studied by M. Gill, and coincides with the Kahler-Ricci flow if the initial metric is Kahler. We find the maximal existence time for the flow in terms of the initial data. We investigate the behavior of the flow on complex surfaces when the initial metric is Gauduchon, on complex manifolds with negative first Chern class, and on some Hopf manifolds. Finally, we discuss a new estimate for the complex Monge-Ampere equation on Hermitian manifolds.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0f4a441e578c0abc8ae7e8acf73652a5041288d8

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Ricci Curvatures on Hermitian manifolds (2014, methodology): “on non-K¨ahler Hermitian manifolds with vanishing ﬁrst Bott-Chern classes. They are always characterized by using the ﬁrst Chern-Ricci curvature Θ(1) and the related Monge-Amp`ere type equations (e.g.[18, 25, 51, 52, 53, 54, 55]). For more details on this subject, we refer the reader to the nice sur”
Scalar curvature on compact complex manifolds (2017, background): “This example is studied with details in [15, 16, 21, 22, 23, 26, 27].”
Non-Kähler Calabi-Yau geometry and pluriclosed flow (2021, background): “[41, 51, 65]), or converging to interesting non-Kähler metrics assuming a certain symmetric ansatz ([45, 50]), Theorem 1.”
The Chern–Ricci flow on complex surfaces (2012, methodology): “ The Chern-Ricci ﬂow is a ﬂow of Hermitian metrics on a complex manifold by their Chern-Ricci form. It was introduced by Gill [21] in the setting of manifolds with vanishing ﬁrst Bott-Chern class. In [58], the authors proved a number of further properties of the Chern-Ricci ﬂow, several of which are”
The Monge–Ampère equation for non-integrable almost complex structures (2016, methodology): “This is in fact equivalent to the evolution equation for almost Hermitian forms ∂ωt ∂t = −Ric(ωt)((1,1)), ω0 = ω, which is the Kähler-Ricci flow if J is integrable and ω0 is Kähler, and the Chern-Ricci flow [23, 49] if J is integrable.”