A parabolic flow of pluriclosed metrics

By Jeffrey Streets, G. Tian. 2009

267 citations recorded by the source corpus.

We define a parabolic flow of pluriclosed metrics. This flow is of the same family introduced by the authors in \cite{ST}. We study the relationship of the existence of the flow and associated static metrics topological information on the underlying complex manifold. Solutions to the static equation are automatically Hermitian-symplectic, a condition we define herein. These static metrics are classified on K3 surfaces, complex toroidal surfaces, nonminimal Hopf surfaces, surfaces of general type, and Class $\seven^+$ surfaces. To finish we discuss how the flow may potentially be used to study the topology of Class $\seven^+$ surfaces.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/698bcc3539fb05bae8f68c943e9f38fa84a841f5

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Six-Dimensional Solvmanifolds with Holomorphically Trivial Canonical Bundle (2014, background): “The pair (ω, J) is also called a Hermitian-symplectic structure in [30].”
Symplectic curvature flow (2010, background): “4) a gradient flow like the Ricci flow and the pluriclosed flow of [19]? We showed in [20] that the parabolic flow of pluriclosed metrics of [19] is in fact a gradient flow.”
Rational curves and special metrics on twistor spaces (2012, methodology): “In [28] Streets and Tian have constructed a parabolic flow of pluriclosed metrics, analogous to the Kähler–Ricci flow.”
Special Hermitian metrics on compact solvmanifolds (2015, methodology): “In [44] it is proved that the Hermitian curvature flow preserves the SKT condition in the sense that when the initial metric go is SKT, then the solution g(t) to (5) holds SKT for every t.”
On Bismut flat manifolds (2016, methodology): “As another example, in [35], Streets and Tian used Bismut connection to reinterpret their Hermitian curvature flow ([33], [34]) and exhibited a remarkable relationship to mathematical physics.”
On Curvature Tensors of Hermitian Manifolds (2016, unknown): “See for instance the work of Fu-Yau [5], Fu-Li-Yau [6], Fu [4], Fu-Wang-Wu [7], [8], Liu-Yang [16], [17], [18], Streets-Tian [20], Tosatti-Weinkove [24], [22], Guan-Sun [13], and the references therein.”
The long‐time behavior of the homogeneous pluriclosed flow (2017, background): “Moreover, if the initial condition ω0 is Kähler both flows coincide (see [30]).”
Strominger connection and pluriclosed metrics (2019, methodology): “We refer the readers to the papers [8], [13], [14], [16], [26], [39] for study involving Hull-Strominger system, which uses Bismut-Strominger connection in an essential way, and to the papers [33], [34], [35] where the study of Hermitian curvature flow involving the connection were initiated which h”
Ricci Curvatures on Hermitian manifolds (2014, methodology): “[45, 46, 47], [39], [24, 51, 52, 55, 25, 26]) by using the well-studied Hamilton’s Ricci flow.”
A correction to “Tamed symplectic forms and strong Kähler with torsion metrics” (2012, methodology): “From [28, 39] there exist no examples in dimension 4.”
Regularity results for pluriclosed flow (2010, methodology): “By the results of [27], the corresponding elliptic (static) equation, ie, (8-1) P(1;1) D !; seems very closely related to the Kähler–Einstein condition, and we further seen here the relationship of solutions to (1-1) and the topological and complex structure of surfaces.”
Geometry of Hermitian manifolds (2010, background): “For more details, we refer the reader to [44].”
The Chern–Ricci flow on complex surfaces (2012, methodology): “Other flows of Hermitian metrics have been previously studied by Streets-Tian [48, 49, 50], motivated in part by the open problem of classifying Class VII surfaces.”
Non-Kähler Calabi-Yau geometry and pluriclosed flow (2021, methodology): “With the goal of constructing canonical geometries on complex manifolds in mind, the third author and Tian introduced a natural geometric flow of pluriclosed metrics, called pluriclosed flow [57].”
On the evolution of a Hermitian metric by its Chern-Ricci form (2011, background): “property have been proposed by Streets-Tian [42, 43] and also Liu-Yang [28].”