Ricci flow of conformally compact metrics

Bahuaud, Eric

doi:10.1016/j.anihpc.2011.03.007

Cited by 18 publications

(55 citation statements)

References 24 publications

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“…is a complete metric with uniformly bounded curvature, and so by the work of Chen-Zhu [4] (see in particular Theorem 2.1 and observe that the estimates allow us to continue a solution to [0, T 1 ]), there exists a solution to the harmonic map heat flow coupled with the Ricci flow on [0, T 1 ] that produces a solution g(t) to the normalized Ricci DeTurck flow on [0, T 1 ] with uniform control of curvature and all covariant derivatives. But now since g(0) = g 0 is conformally compact asymptotically hyperbolic the regularity results in [2,Section 5] give that g(t) is smoothly conformally compact AH on [0, T 1 ]. Pullback by the DeTurck diffeomorphism now extends the conformally compact AH solution of the NRF on [0, T 1 ], contradicting the maximality of T 1 .…”

Section: 2mentioning

confidence: 99%

Asymptotically hyperbolic normalized Ricci flow and rotational symmetry

Bahuaud

Woolgar

2018

Communications in Analysis and Geometry

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We consider the normalized Ricci flow evolving from an initial metric which is conformally compactifiable and asymptotically hyperbolic. We show that there is a unique evolving metric which remains in this class, and that the flow exists up to the time where the norm of the Riemann tensor diverges. Restricting to initial metrics which belong to this class and are rotationally symmetric, we prove that if the sectional curvature in planes tangent to the orbits of symmetry is initially nonpositive, the flow starting from such an initial metric exists for all time. Moreover, if the sectional curvature in planes tangent to these orbits is initially negative, the flow converges at an exponential rate to standard hyperbolic space. This restriction on sectional curvature automatically rules out initial data admitting a minimal hypersphere.

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Section: 2mentioning

confidence: 99%

Asymptotically hyperbolic normalized Ricci flow and rotational symmetry

Bahuaud

Woolgar

2018

Communications in Analysis and Geometry

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“…This makes an initially zero mass become negative, contradicting the positive energy theorem for asymptotically hyperbolic metrics with round sphere conformal infinity [38,13]. The perturbation constructed in [4] was essentially an infinitesimal version of a normalized Ricci flow (this flow was studied by [8]). But the flow of [8] will change the mass if the mass is not zero before the perturbation [11], and will cause a negative mass to increase (to become closer to zero), so this process does not necessarily produce a net decrease in mass.…”

Section: 5mentioning

confidence: 99%

“…The perturbation constructed in [4] was essentially an infinitesimal version of a normalized Ricci flow (this flow was studied by [8]). But the flow of [8] will change the mass if the mass is not zero before the perturbation [11], and will cause a negative mass to increase (to become closer to zero), so this process does not necessarily produce a net decrease in mass. That is why a naïve application of the [4] approach will not work.…”

Section: 5mentioning

confidence: 99%

The rigid Horowitz-Myers conjecture

Woolgar

2017

J. High Energ. Phys.

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Abstract:The new positive energy conjecture was first formulated by Horowitz and Myers in the late 1990s to probe for a possible extended, nonsupersymmetric AdS/CFT correspondence. We consider a version formulated for complete, asymptotically Poincaré-Einstein Riemannian metrics (M, g) with bounded scalar curvature R ≥ −n(n − 1) and with no (inner) boundary except possibly a finite union of compact, totally geodesic hypersurfaces (horizons). This version then asserts that any such (M, g) must have mass not less than a certain bound which is realized as the mass m 0 of a metric g 0 induced on a time-symmetric slice of a spacetime called an AdS soliton. This conjecture remains unproved, having so far resisted standard techniques. Little is known other than that the conjecture is true for metrics which are sufficiently small perturbations of g 0 . We pose another test for the conjecture. We assume its validity and attempt to prove as a corollary the corresponding scalar curvature rigidity statement, which is that g 0 is the unique asymptotically Poincaré-Einstein metric with mass m 0 obeying R ≥ −n(n − 1). Were a second such metric g 1 not isometric to g 0 to exist, it then may well admit perturbations of lower mass, contradicting the assumed validity of the conjecture. We find enough rigidity to show that the minimum mass metric must be static Einstein, so the problem is reduced to that of static uniqueness. When n = 3 the manifold must be isometric to a time-symmetric slice of an AdS soliton spacetime, or must have a non-compact horizon. En route we study the mass aspect, obtaining and generalizing known results: (i) we relate the mass aspect of static metrics to the holographic energy density, (ii) we obtain the conformal invariance of the mass aspect when the bulk dimension is odd, and (iii) we show the vanishing of the mass aspect for negative Einstein manifolds with Einstein conformal boundary.

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“…, t ∈ [0, ∞) be a solution to the NRDF (2). Then there exist constantsǫ,ǭ > 0, depending only on ǫ, n andg such that…”

Section: Proof Of the Volume Comparisonmentioning

confidence: 99%

“…where the components of V are given by V i := g ij V j , then we obtain a family of smooth diffeomorphisms Φ t for t > 0 such that if g(t), t ∈ [0, T ) is a solution to NRDF (2),ḡ(t) := Φ * t g(t), t ∈ [0, T ) is a solution to NRF (1). There are several papers which investigated the stability of hyperbolic space under NRF [14,20,2,3]. In [20], Schnürer, Schulze and Simon used the NRDF to get the stability of hyperbolic space.…”

Section: Introductionmentioning

confidence: 99%

Volume Comparison of Conformally Compact Manifolds with Scalar Curvature R ≥ −n (n − 1)

Han

Shi

2015

Ann. Henri Poincaré

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In this paper, we use the normalized Ricci-DeTurk flow to prove a stability result for strictly stable conformally compact Einstein manifolds. As an application, we show a local volume comparison of conformally compact manifolds with scalar curvature R ≥ −n (n − 1) and also the rigidity result when certain relative volume is zero.Resumé. Dans cet article, nous utilisons le flot de Ricci-DeTurk normalisé pour prouver la stabilité des variétés d'Einstein strictement stables et conformément compactes. En tant qu'application, nous montrons une comparaison de volume local pour les variétés conformément compactes dont la courbure scalaire satisfait R ≥ −n(n − 1). Nous donnonségalement un résultat de rigidité lorsque certain volume relative est nul.

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Ricci flow of conformally compact metrics

Cited by 18 publications

References 24 publications

Asymptotically hyperbolic normalized Ricci flow and rotational symmetry

Asymptotically hyperbolic normalized Ricci flow and rotational symmetry

The rigid Horowitz-Myers conjecture

Volume Comparison of Conformally Compact Manifolds with Scalar Curvature R ≥ −n (n − 1)

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