Short-time persistence of bounded curvature under the Ricci flow

Kotschwar, Brett

doi:10.4310/mrl.2017.v24.n2.a9

Cited by 6 publications

(6 citation statements)

References 10 publications

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“…is quadratic in h and its derivatives. Next, we recall the evolution equation satisfied by the Bianchi gauge along the Ricci flow with a Ricci flat background metric g ∞ given in [Kot17]: Lemma 3.11 (Kotschwar). Let (N n , g ∞ ) be a Ricci flat metric and let (g(t)) t∈[0,T ) be a Ricci flow on N .…”

Section: Proof Of Theorem 31mentioning

confidence: 99%

Dynamical (in)stability of Ricci-flat ALE metrics along Ricci flow

Deruelle¹,

Ozuch²

2021

Preprint

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We study the stability and instability of ALE Ricci-flat metrics around which a Lojasiewicz inequality is satisfied for a variation of Perelman's λ functional adapted to the ALE situation and denoted λ ALE . This functional was introduced by the authors in a recent work and it has been proven that it satisfies a good enough Lojasiewicz inequality at least in neighborhoods of integrable ALE Ricci-flat metrics in dimension larger than or equal to 5. Contents 1.2. The functionals λ 0 ALE , m ADM and λ ALE 1.3. Main properties of λ ALE 2. Preliminary estimates for short and large time 2.1. Heat kernel Gaussian bounds 2.2. L 2 − C 0 estimate 2.3. Weighted estimates 3. Stability of Ricci-flat ALE metrics 3.1. A stability result 3.2. Proof of Theorem 3.1 3.3. Evolution of the Bianchi form, the scalar curvature and the mass 3.4. A discussion on previous stability results 4. Instability of Ricci-flat ALE metrics 4.1. A priori energy estimates in the unstable case 4.2. A digression on λ ALE 4.3. An unstability result Appendix A. First and second variations of geometric quantities References Date: today.

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Section: Proof Of Theorem 31mentioning

confidence: 99%

Dynamical (in)stability of Ricci-flat ALE metrics along Ricci flow

Deruelle¹,

Ozuch²

2021

Preprint

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“…In [18], the author consider the uniqueness problem if the curvature is bounded above by Ct −γ for γ ∈ [0, 1). By integrability of the Ricci curvature, two solutions g(t) and g(t) are also uniform equivalent to g 0 and hence to each other.…”

Section: Uniqueness Of Kähler-ricci Flowmentioning

confidence: 99%

“…When n = 3, Chen [5] obtained a strong uniqueness result which says that any complete solution of the Ricci flow starting from the Euclidean metric must be stationary. Recently, Kotschwar [19,18] introduced an energy method which extended the classical uniqueness result to the case when two flows are uniformly equivalent and their curvature is bounded above by C(d 0 (x, p) 2 + 1)/t γ , γ < 1/2 or C/t γ , γ < 1. However, most of the solutions mentioned in the constructions above satisfy a curvature bound of the form C/t.…”

Section: Introductionmentioning

confidence: 99%

On the Uniqueness of Ricci Flow

Lee

2018

J Geom Anal

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“…The energy method can also be used in proving uniqueness of other types of geometric flows. For example, Kotschwar applied the energy method to prove the uniqueness of Ricci flow [13,14]. Part of our motivation of the current work arises from our study of another Schrödinger type geometric flow, namely, the Skew Mean Curvature Flow(SMCF) [20].…”

Section: Introductionmentioning

confidence: 99%