Stability of Einstein metrics under Ricci flow

Kroencke, Klaus

doi:10.4310/cag.2020.v28.n2.a5

Cited by 25 publications

(25 citation statements)

References 19 publications

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“…Here the metric on M × I is given by g s + ds 2 provided that g s is chosen such that div g s 0 d ds | s=s 0 g s = 0 and the spinor is given by ( 22) resp. (23). In particular the spinor can be normalized such that it has norm F(t) at any (x, t) ∈ M × I .…”

Section: Remark 13 In This Example We Have Seen Two Different Ways Of Reconstructing Lorentzianmentioning

confidence: 99%

“…Parallel spinors are also linked to "stability", defined in the sense that the given compact Ricci-flat metric cannot be deformed to a positive scalar curvature metric. This condition in turn is linked to dynamical stability of a Ricci-flat metric under Ricci flow: A compact Ricci-flat metric is dynamically stable under the Ricci flow if and only if it cannot be deformed to a positive scalar curvature metric ( [20] and [23,Theorem 1.1]).…”

Section: Introductionmentioning

confidence: 99%

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Construction of Initial Data Sets for Lorentzian Manifolds with Lightlike Parallel Spinors

Ammann

Kröncke

Müller

2021

Commun. Math. Phys.

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Lorentzian manifolds with parallel spinors are important objects of study in several branches of geometry, analysis and mathematical physics. Their Cauchy problem has recently been discussed by Baum, Leistner and Lischewski, who proved that the problem locally has a unique solution up to diffeomorphisms, provided that the intial data given on a space-like hypersurface satisfy some constraint equations. In this article we provide a method to solve these constraint equations. In particular, any curve (resp. closed curve) in the moduli space of Riemannian metrics on M with a parallel spinor gives rise to a solution of the constraint equations on $$M\times (a,b)$$ M × ( a , b ) (resp. $$M\times S^1$$ M × S 1 ).

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Section: Remark 13 In This Example We Have Seen Two Different Ways Of Reconstructing Lorentzianmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Construction of Initial Data Sets for Lorentzian Manifolds with Lightlike Parallel Spinors

Ammann

Kröncke

Müller

2021

Commun. Math. Phys.

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“…In fact, the conformal variations are built from the eigenfunctions associated to the first non‐zero eigenvalue of the ordinary Laplacian. Recently, Kröncke proved a stability criterion that can be checked for such neutral conformal directions . The criterion simply involves integrating the cube of the eigenfunctions and determining if the resulting integral is non‐zero (see Theorem ).…”

Section: Introductionmentioning

confidence: 99%

The canonical Einstein metric on G2 is dynamically unstable under the Ricci flow

Hall

2019

Bull. London Math. Soc.

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“…The relationship between infinitesimal deformations of Einstein metrics, their integrability, and the dynamical stability of the Ricci flow has been known for quite some time through the works of Sesum [30], Haslhofer and Müller [12], and Kröncke [17]. In Section 6 we compute the third variation of Perelman's ν-functional for infinitesimal deformations of an Einstein metric (this is related to the calculation made for conformal variations by Kröncke [20] -see also the calculation of Knopf and Sesum [13]). The next theorem shows that the variation is a multiple of Koiso's obstruction to integrability I (see Section 2 for the precise definition of this obstruction).…”

mentioning

confidence: 99%

“…(see [11], [13], or [20]). Hence the obstruction is a non-zero multiple of Ψ and so does not vanish for any perturbation induced by v ∈ V −2λ .…”

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confidence: 99%

Rigidity of $SU_n$-type symmetric spaces

Batat¹,

Hall²,

Murphy³

et al. 2021

Preprint

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We prove that the bi-invariant Einstein metric on SU2n+1 is isolated in the moduli space of Einstein metrics, even though it admits infinitesimal deformations. This gives a non-Kähler, non-product example of this phenomenon adding to the famous example of CP 2n × CP 1 found by Koiso. We apply our methods to derive similar solitonic rigidity results for the Kähler-Einstein metrics on 'odd' Grassmannians. We also make explicit a connection between non-integrable deformations and the dynamical instability of metrics under Ricci flow.

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Stability of Einstein metrics under Ricci flow

Cited by 25 publications

References 19 publications

Construction of Initial Data Sets for Lorentzian Manifolds with Lightlike Parallel Spinors

Construction of Initial Data Sets for Lorentzian Manifolds with Lightlike Parallel Spinors

The canonical Einstein metric on G2 is dynamically unstable under the Ricci flow

Rigidity of $SU_n$-type symmetric spaces

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