Ricci-Yang-Mills flow on surfaces and pluriclosed flow on elliptic fibrations

Streets, Jeffrey

doi:10.48550/arxiv.2102.09538

Cited by 2 publications

(3 citation statements)

References 18 publications

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“…Furthermore, assuming the solution satisfies type III curvature and diameter bounds, this kind of limiting behavior can be derived by the use of an expanding entropy functional [34]. Assuming the initial metric is T 2 -invariant this behavior was shown in [50].…”

Section: Pluriclosed Flowmentioning

confidence: 99%

“…From a more geometric point of view, we note that while recently there have been some results on geometric flows of non-Kähler metrics converging to rigid, Kähler metrics (cf. [41,51,65]), or converging to interesting non-Kähler metrics assuming a certain symmetric ansatz ( [45,50]), Theorem 1.2 seems to be the first result showing that a natural class of non-Kähler metrics is globally attractive for a geometric flow with arbitrary initial data.…”

Section: In View Of Our Description Of Pluriclosed Flow In Terms Of M...mentioning

confidence: 99%

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Non-Kähler Calabi-Yau geometry and pluriclosed flow

Garcia-Fernandez¹,

Jordan²,

Streets³

2021

Preprint

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Hermitian, pluriclosed metrics with vanishing Bismut-Ricci form give a natural extension of Calabi-Yau metrics to the setting of complex, non-Kähler manifolds, and arise independently in mathematical physics. We reinterpret this condition in terms of the Hermitian-Einstein equation on an associated holomorphic Courant algebroid, and thus refer to solutions as Bismut Hermitian-Einstein. This implies Mumford-Takemoto slope stability obstructions, and using these we exhibit infinitely many topologically distinct complex manifolds in every dimension with vanishing first Chern class which do not admit Bismut Hermitian-Einstein metrics. This reformulation also leads to a new description of pluriclosed flow in terms of Hermitian metrics on holomorphic Courant algebroids, implying new global existence results, in particular on all complex non-Kähler surfaces of Kodaira dimension κ ≥ 0. On complex manifolds which admit Bismut-flat metrics we show global existence and convergence of pluriclosed flow to a Bismut-flat metric, which in turn gives a classification of generalized Kähler structures on these spaces.

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Section: Pluriclosed Flowmentioning

confidence: 99%

Section: In View Of Our Description Of Pluriclosed Flow In Terms Of M...mentioning

confidence: 99%

Non-Kähler Calabi-Yau geometry and pluriclosed flow

Garcia-Fernandez¹,

Jordan²,

Streets³

2021

Preprint

Self Cite

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show abstract

“…Related Flows. We want to raise similarities between this flow and 2 other flows found in literature: Ricci-Yang-Mills flow by Jeffrey Streets [Str07,Str21] and hypersymplectic flow by Fine and Yao [FY20].…”

Section: Ricci Flow On Principal G-bundlesmentioning

confidence: 83%

Ricci Flow on Torus Bundles

Yeroshkin

2021

Preprint

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In this paper we compute the Ricci flow formulas for invariant metrics on prinicpal G-bundles compatible with the connection. Our primary focus is on torus bundles which we use to study a notion of Bakry-Émery Ricci flow as well as Ricci flow on circle bundles over Kähler-Einstein manifolds. The latter application gives us solutions to Ricci flows on Heisenberg groups and implicit solutions to Ricci flows on Berger 3-spheres and several other 3-dimensional manifolds.

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Ricci-Yang-Mills flow on surfaces and pluriclosed flow on elliptic fibrations

Abstract: We give a complete description of the global existence and convergence for the Ricci-Yang-Mills flow on T k bundles over Riemann surfaces. These results equivalently describe solutions to generalized Ricci flow and pluriclosed flow with symmetry.

Cited by 2 publications

References 18 publications

Non-Kähler Calabi-Yau geometry and pluriclosed flow

Non-Kähler Calabi-Yau geometry and pluriclosed flow

Ricci Flow on Torus Bundles

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