Hermitian curvature flow

Streets, Jeffrey; Тиан, Ганг

doi:10.4171/jems/262

Cited by 126 publications

(139 citation statements)

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“…The following result indicates that the parabolic ow (AHCF) could play the same role as the ow (HCF) Q on complex manifolds. We may expect to have some other similar results as in [5], [6] and [7] for (AHCF). Theorem 1.2.…”

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confidence: 73%

An almost pluriclosed flow

Kawamura¹

2017

Geometric Flows

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Abstract:We de ne two parabolic ows on almost complex manifolds, which coincide with the pluriclosed ow and the Hermitian curvature ow respectively on complex manifolds. We study the relationship between these parabolic evolution equations on compact almost Hermitian manifolds.Keywords: almost Hermitian metric, pluriclosed ow, Hermitian curvature ow MSC: 53C44 (primary); 53C55; 32W20 (secondary) IntroductionIn [5] and [7], Streets and Tian introduced a parabolic evolution equation of pluriclosed metrics with a pluriclosed initial metric on a Hermitian manifold, which is called the pluriclosed ow. In this paper, we would like to show that some of their results hold for almost Hermitian cases as well. Let (M, J) be a compact almost complex manifold and let g be an almost Hermitian metric on M. Let {Zr} be an arbitrary local ( , )-frame around a xed point p ∈ M and let {ζ r } be the associated coframe. Then the associated real ( , )-form ω with respect to g takes the local expression ω = √ − g rk ζ r ∧ ζk. We will also refer to ω as to an almost Hermitian metric. We would like to de ne a parabolic ow of almost Hermitian metrics with an almost pluriclosed initial metric ω on (M, J). We say a metric ω is almost pluriclosed if ω is an almost Hermitian metric and ∂∂-closed (cf. De nition 1.1). We will call it the almost pluriclosed ow (APF): g(t) are the L -adjoint operators with respect to metrics g(t), and P(ω) is one of the Riccitype curvatures of the Chern curvature. One has with an arbitrary ( , )-frame {Zr} with respect to g, P ij =where Ω is the curvature of the Chern connection ∇ on (M, g, J) and O(∂ω) means an expression which only depends on at most rst derivatives of ω. The rst goal of this paper is to prove that the operator ω → Φ(ω) is a strictly elliptic operator for an almost pluriclosed metric ω, which means that the equation (APF) with an almost pluriclosed initial metric is a strictly parabolic equation. Hence the short-time existence and the uniqueness of the solution (APF) follows from the standard parabolic theory since the manifold is supposed to be compact. This ow (APF) coincides with the pluriclosed ow if J is integrable and also the initial metric is pluriclosed (cf. 73We will also refer to the associated real ( , )-form ω as an almost pluriclosed metric. Theorem 1.1. Given a compact almost Hermitian manifold (M, ω , J) with almost pluriclosed metric ω , there exists a unique solution ω(t) to (APF) with initial condition ω for t ∈ [ , ε) for some ε > . Moreover, if ω( ) is almost pluriclosed, the metric ω(t) is almost pluriclosed for all t ∈ [ , ε).We denote by S one of the Ricci-type curvatures of the Chern curvature, which is locally given by S ij = g kl Ω klij. The second goal of this paper is to prove that a solution of the almost pluriclosed ow with initial almost pluriclosed metric ω is equivalent to a solution of the following parabolic ow starting at the initial almost pluriclosed metric ω on a compact almost complex manifold with an almost pluriclosed Hermitian metric, we will call...

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confidence: 73%

An almost pluriclosed flow

Kawamura¹

2017

Geometric Flows

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“…One can show general short-time existence for (5) as well as smoothing estimates, generalizing Theorem 1.1 of [7]. Of course if the initial complex structure is integrable, then the resulting evolution fixes J , and one is reduced to a flow of Hermitian metrics.…”

Section: Theorem 2 (See [8 Theorem 13]) Let (M 2n G J ) Be a Cmentioning

confidence: 96%

“…This viewpoint was used in [7] and [6] to prove certain basic regularity theorems for (1), and indeed a wider class of flows of Hermitian metrics. Observe here that S is the curvature term appearing in Hermitian Yang-Mills theory on the tangent bundle, the only important difference being that we are not taking the trace with respect to a fixed background metric, but rather the given Hermitian metric.…”

mentioning

confidence: 98%

Regularity theory for pluriclosed flow

Streets

Тиан

2010

Comptes Rendus. Mathématique

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“…This viewpoint was considered recently by Vezzoni [25]. When J is integrable, this is precisely the family of equations introduced in [18]. If one is interested in understanding metrics compatible with a given almost complex structure, (1.3) could be a useful tool.…”

Section: Introductionmentioning

confidence: 97%

“…Remark 1.3. When J 0 is integrable, the one-parameter family of metrics !.t / is a solution to Hermitian curvature flow, as defined in [18]. Again, the torsion term Q can be arbitrary for the result of Theorem 1.1.…”

Section: Introductionmentioning

confidence: 97%

Symplectic curvature flow

Streets

Тиан

2013

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We introduce a parabolic flow of almost Kähler structures, providing an approach to constructing canonical geometric structures on symplectic manifolds. We exhibit this flow as one of a family of parabolic flows of almost Hermitian structures, generalizing our previous work on parabolic flows of Hermitian metrics. We exhibit a long time existence obstruction for solutions to this flow by showing certain smoothing estimates for the curvature and torsion. We end with a discussion of the limiting objects as well as some open problems related to the symplectic curvature flow.

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Hermitian curvature flow

Abstract: Abstract. We define a functional for Hermitian metrics using the curvature of the Chern connection. The Euler-Lagrange equation for this functional is an elliptic equation for Hermitian metrics.

Cited by 126 publications

References 13 publications

An almost pluriclosed flow

An almost pluriclosed flow

Regularity theory for pluriclosed flow

Symplectic curvature flow

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