Long time existence and bounded scalar curvature in the Ricci-harmonic flow

Li, Yang

doi:10.1016/j.jde.2018.02.028

Cited by 13 publications

(17 citation statements)

References 56 publications

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“…Here we list some results both for the Ricciharmonic flow and the Ricci flow, see Table 1. Besides these results, there are other works on the Ricci-harmonic flow including gradient estimates, eigenvalues, entropies, functionals, and solitons, etc., see [8,13,18,19,20,21,43,44,50].…”

Section: Introductionmentioning

confidence: 99%

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Local curvature estimates for the Ricci-harmonic flow

Li¹

2018

Preprint

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In this paper we give an explicit bound of ∆ g(t) u(t) and the local curvature estimates for the Ricci-harmonic flowunder the condition that the Ricci curvature is bounded along the flow. In the second part these local curvature estimates are extended to a class of generalized Ricci flow, introduced by the author [42], whose stable points give Ricci-flat metrics on a complete manifold, and which is very close to the (K, N)-super Ricci flow recently defined by Xiangdong Li and Songzi Li [35]. Next we propose a conjecture for Einstein's scalar field equations motivated by a result in the first part and the bounded L 2 -curvature conjecture recently solved by Klainerman, Rodnianski and Szeftel [26]. In the last two parts of this paper, we discuss two notions of "Riemann curvature tensor" in the sense of 24,67,68], respectively, and Li [44], whose "Ricci curvature" both give the standard Bakey-Émery Ricci curvature [1], and the forward and backward uniqueness for the Ricci-harmonic flow. CONTENTS1. Introduction 2. Gradient and local curvature estimates 2.1. The boundedness of ∆ g(t) u(t) 2.2. Local curvature estimates 3. Results for a generalized Ricci flow 3.1. Long time existence 3.2. Bounded scalar curvature 4. Bounded L 2 -curvature conjecture for the Einstein scalar field equations 4.1. Initial value problem 4.2. Bounded L 2 -curvature conjecture for Einstein's equations 4.3. Bounded L 2 -curvature conjecture for the Einstein scalar field equations 5. Sm and Wylie-Yeroshkin Riemann curvature 5.1. Integral inequalities for scalar and Ricci curvatures 5.2. Killing vector fields with constant length 5.3. Remark on Rm L and Rm WY 6. Uniqueness for the Ricci-harmonic flow 6.1. Forward uniqueness 6.2. Backward uniqueness Appendix A. Evolution equations of the Ricci-harmonic flow 2010 Mathematics Subject Classification. Primary 53C44.

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Section: Introductionmentioning

confidence: 99%

“…(B) Results for a generalized Ricci flow. The author [43] introduced a class of generalized Ricci flow (For motivation see Section 3), called (α 1 , 0, β 1 , β 2 )-Ricci flow:…”

Section: Introductionmentioning

confidence: 99%

Local curvature estimates for the Ricci-harmonic flow

Li¹

2018

Preprint

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“…For the Ricci-harmonic flow introduce by List [30,31] (see also, [35,36]), the analogue of Theorem 1.2 was proved in [30,31] (see also, [35,36]) and [4] (see [28] for another proof). The author [26,27] extended Cao's result [3] to the Ricci-harmonic flow. The same Hamilton's conjecture was asked by the author in [26,27].…”

Section: Theorem 13mentioning

confidence: 88%

“…The author [26,27] extended Cao's result [3] to the Ricci-harmonic flow. The same Hamilton's conjecture was asked by the author in [26,27].…”

Section: Theorem 13mentioning

confidence: 88%

Local curvature estimates for the Laplacian flow

2021

Calc. Var.

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In this paper we give local curvature estimates for the Laplacian flow on closed $$G_{2}$$ G 2 -structures under the condition that the Ricci curvature is bounded along the flow. The main ingredient consists of the idea of Kotschwar et al. (J Funct Anal 271(9):2604–2630, 2016) who gave local curvature estimates for the Ricci flow on complete manifolds and then provided a new elementary proof of Sesum’s result (Sesum in Am J Math 127(6):1315–1324, 2005), and the particular structure of the Laplacian flow on closed $$G_{2}$$ G 2 -structures. As an immediate consequence, this estimates give a new proof of Lotay and Wei’s (Geom Funct Anal 27(1):165–233, 2017) result which is an analogue of Sesum’s theorem. The second result is about an interesting evolution equation for the scalar curvature of the Laplacian flow of closed $$G_{2}$$ G 2 -structures. Roughly speaking, we can prove that the time derivative of the scalar curvature $$R_{g(t)}$$ R g ( t ) is equal to the Laplacian of $$R_{g(t)}$$ R g ( t ) , plus an extra term which can be written as the difference of two nonnegative quantities.

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“…On the other hand, those curvatures are L 2 bounded in certain cases (e.g. n = 4) if scalar curvature is bounded (see [18]). Furthermore, the pseudo-locality theorem corresponding to the Ricci-harmonic flow was be given in [9].…”

Section: Introductionmentioning

confidence: 99%

A local curvature estimate for the Ricci-harmonic flow on complete Riemannian manifolds

Li¹,

Zhang²

2021

Preprint

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In this paper we consider the local L p estimate of Riemannian curvature for the Ricci-harmonic flow or List's flow introduced by List [21] on complete noncompact manifolds. As an application, under the assumption that the flow exists on a finite time interval [0, T) and the Ricci curvature is uniformly bounded, we prove that the L p norm of Riemannian curvature is bounded, and then, applying the De Giorgi-Nash-Moser iteration method, obtain the local boundedness of Riemannian curvature and consequently the flow can be continuously extended past T.

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Long time existence and bounded scalar curvature in the Ricci-harmonic flow

Cited by 13 publications

References 56 publications

Local curvature estimates for the Ricci-harmonic flow

Local curvature estimates for the Ricci-harmonic flow

Local curvature estimates for the Laplacian flow

A local curvature estimate for the Ricci-harmonic flow on complete Riemannian manifolds

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