Second-Order Geometric Flows on Foliated Manifolds

Bedulli, Lucio; He, Weiyong; Vezzoni, Luigi

doi:10.1007/s12220-017-9839-7

Cited by 7 publications

(8 citation statements)

References 28 publications

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“…Alternatively, the short-time existence of the Sasaki J-flow can be obtained by invoking the short-time existence of any second order transversally parabolic equation on compact manifolds foliated by Riemannian foliations. A proof of the latter result can be founded in [1].…”

Section: Well-posedness Of the Sasaki J-flowmentioning

confidence: 87%

On the J-flow in Sasakian manifolds

Vezzoni

Zedda

2015

Annali di Matematica

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We study the space of Sasaki metrics on a compact manifold M by introducing an odddimensional analogue of the J-flow. That leads to the notion of critical metric in the Sasakian context. In analogy to the Kähler case, on a polarised Sasakian manifold there exists at most one normalised critical metric. The flow is a tool for texting the existence of such a metric. We show that some results proved by Chen in [7] can be generalised to the Sasakian case. In particular, the Sasaki J-flow is a gradient flow which has always a long-time solution minimising the distance on the space of Sasakian potentials of a polarized Sasakian manifold. The flow minimises an energy functional whose definition depends on the choice of a background transverse Kähler form χ. When χ has nonnegative transverse holomorphic bisectional curvature, the flow converges to a critical Sasakian structure.

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Section: Well-posedness Of the Sasaki J-flowmentioning

confidence: 87%

On the J-flow in Sasakian manifolds

Vezzoni

Zedda

2015

Annali di Matematica

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“…Proof. The proof simply follows from Theorem 4 in [41] and Theorem 5.1 in [12]. Since our totally geodesic foliation structure satisfies the homologically oriented Riemannian foliation condition and our transverse metric g H (t) is a smooth holonomy invariant metric on the quotient bundle.…”

Section: Structure Under Transverse Ricci Flowmentioning

confidence: 92%

“…There are several recent study of Ricci flow on Riemannian foliations. For example: mixed curvature Ricci flow on co-dimension one foliations [41,42], Sasakian transverse Ricci flow [15,43], general second order geometric flows on Riemannian foliations [12]. But in all these previous work, they all have torsion free condition for the transverse Levi-Civita connection.…”

Section: Introductionmentioning

confidence: 99%

Harnack inequalities on totally geodesic foliations with transverse Ricci flow

Feng¹

2017

Preprint

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In the current paper,under the transverse Ricci flow on a totally geodesic Riemannian foliation, we prove two types of differential Harnack inequalities (Li-Yau gradient estimate) for the positive solutions of the heat equation associated with the time dependent horizontal Laplacian operators. We also get a time dependent version of the generalized curvature dimension inequality. As a consequence of aforementioned results, we also get parabolic Harnack inequalities and heat kernel upper bounds.

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“…In the joint work with L. Vezzoni [23], we study the long time existence of the J-flow (2) and prove its convergence to a critical metric under an additional hypothesis on the sign of the transverse holomorphic sectional curvature of χ. In the recent paper [1], the Sasaki J-flow is included as particular case in a more general result that prove the short time existence of second order geometric flows on foliated manifolds.…”

Section: Sasakian Manifolds and The Sasaki J-flowmentioning

confidence: 99%

“…Normalize χ in order to get c = 1 (i.e. 1 2 dη − (n − 1)χ > 0). Fix t > 0 and let (x 0 , t 0 ) be a maximum in M × [0, t] for log γ f − Af , where A is a constant to be fix later.…”

Section: Second Order Estimatesmentioning

confidence: 99%