Rigidity of conformally compact manifolds with the round sphere as the conformal infinity

Dutta, Satyaki; Javaheri, Mohammad

doi:10.1016/j.aim.2009.12.004

Cited by 21 publications

(34 citation statements)

References 10 publications

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“…Those readily imply the estimates for the curvature of the compactified metric g with the estimates (8), (9) and (10).…”

Section: Basic Estimatesmentioning

confidence: 56%

“…where R is the scalar curvature. We will use our curvature estimates and regularity theorems to replace the C 2 regularity assumption in [10] to prove Theorem 1.6. Since the proof follows the approach in [10] with a number of modifications, we will sketch a proof in the following for readers ′ conveniences.…”

Section: Rigidity Theoremsmentioning

confidence: 99%

“…Although many interesting rigidity theorems were obtained (e.g. [1], [6], [5], [10] [15] [17]) lately, most of them need to assume some regularity of the comformally compactness of manifolds at infinity. As consequences of our regularity theorems we will state and prove Theorem 5.1 and Theorem 5.3 in §5.…”

Section: Introductionmentioning

confidence: 99%

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Regularity and rigidity of asymptotically hyperbolic manifolds

Han

Qing

Shi

2012

Advances in Mathematics

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In this paper, we study some intrinsic characterization of conformally compact manifolds. We show that, if a complete Riemannian manifold admits an essential set and its curvature tends to −1 at infinity in certain rate, then it is conformally compactifiable and the compactified metrics can enjoy some regularity at infinity. As consequences we prove some rigidity theorems for complete manifolds whose curvature tends to the hyperbolic one in a rate greater than 2.

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“…Those readily imply the estimates for the curvature of the compactified metric g with the estimates (8), (9) and (10).…”

Section: Basic Estimatesmentioning

confidence: 56%

Section: Rigidity Theoremsmentioning

confidence: 99%

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Regularity and rigidity of asymptotically hyperbolic manifolds

Han

Qing

Shi

2012

Advances in Mathematics

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“…By the interior estimates of the second order elliptic equations (2.4) − (2.7) and the inequality (3.12), there exists some constant C = C(δ 0 ) > 0 independent of the initial data and the solution so that 3 4 ]. To get global estimates, we multiply x −5 (1 − x 2 ) 6 K 1 2 to (2.5) and do integration on the interval [x, 3 4 ] to obtain 12 for x ∈ (0, 3 4 ], and hence 3 4 ], with some constant C = C(δ 0 ) > 0 independent of the initial data and the solution. Similarly, by multiplying x −2 (1 − x 2 ) 3 K 1 2 to (2.6) and (2.7) and doing integration on [x, 3 4 ] correspondingly, we have 3 4 ), with some constant C = C(δ 0 ) > 0 independent of the solution and the initial data.…”

Section: Uniqueness Of Non-positively Curved Conformally Compact Einsmentioning

confidence: 99%

“…And for K 1 (0) = K 2 (0), by the mean value theorem, there exists a zero of z ′ 1 in x ∈ (0, 1), and Theorem 5.4 in [26] covers this case.) Notice that when φ(0) = 1, the conformal infinity is the conformal class of the round sphere, and by [3][33] [12] [27], the CCE manifold is isometric to the hyperbolic space. Now we are ready to prove the uniqueness result Theorem 1.3.…”

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confidence: 99%

On uniqueness of conformally compact Einstein metrics with homogeneous conformal infinity

2018

Advances in Mathematics

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In this paper we show that for a generalized Berger metricĝ on S 3 close to the round metric, the conformally compact Einstein (CCE) manifold (M, g) with (S 3 , [ĝ]) as its conformal infinity is unique up to isometries. For the high-dimensional case, we show that ifĝ is an SU(k + 1)-invariant metric on S 2k+1 for k ≥ 1, the non-positively curved CCE metric on the (2k + 1)-ball B 1 (0) with (S 2k+1 , [ĝ]) as its conformal infinity is unique up to isometries. In particular, since in [27], we proved that if the Yamabe constant of the conformal infinity Y(S 2k+1 , [ĝ]) is close to that of the round sphere then any CCE manifold filled in must be negatively curved and simply connected, therefore ifĝ is an SU(k + 1)-invariant metric on S 2k+1 which is close to the round metric, the CCE metric filled in is unique up to isometries. Using the continuity method, we prove an existence result of the non-positively curved CCE metric with prescribed conformal infinity (S 2k+1 , [ĝ]) when the metricĝ is SU(k + 1)-invariant. 1 n+1 < λ 1 λ 2 < n + 1, then up to isometry there exists at most one non-positively curved conformally 3 compact Einstein metric on the (n + 1)-ball B 1 (0) with (S n , [ĝ]) as its conformal infinity. In particular, it is the perturbation metric in [16] when λ 1 λ 2 is close to 1. Moreover, by Theorem 2.2 (see also [27]), for λ 1 λ 2 close enough to 1, any conformally compact Einstein manifold filled in is automatically negatively curved and simply connected, and therefore it is unique up to isometry.After that we consider the existence result. Recall that for given real analytic data i.e., the conformal infinity (∂M, [ĝ]) and the non-local term in the expansion in Theorem 2.3, existence of CCE metrics in a neighborhood of the boundary ∂M is proved by Fefferman and Graham in [14] for ∂M of odd dimension and by Kichenassamy in [23] for even dimensional boundary, and for C ∞ data at conformal infinity, see Gursky and Székelyhidi [19]. Anderson [2] studied the existence of CCE metrics on B 4 1 with general prescribed conformal infinity (S 3 , [ĝ]) using the continuity method. Recently, Gursky and Han [18] showed that there are infinitely many Riemannian metricsĝ on S 7 lying in different connected components of the set of positive scalar curvature metrics such that there exists no CCE metrics on the unit Euclidean ball B 8 with (S 7 , [ĝ]) as its conformal infinity and pointed out that similar phenomena holds for higher dimensions.By [26], the CCE manifold which is Hadamard with homogeneous conformal infinity, is of cohomogeneity one. Calculations of the curvature tensors on manifolds of cohomogeneity one can be found in [17]. Recently, using Schauder degree theory, Buttsworth [8] showed that for two G-invariant Riemannian metricsĝ 1 andĝ 2 on a compact homogeneous space G/H, if the isotropy representation of G/H consists of pairwise inequivalent irreducible summands, then there exists an Einstein metric on G/H × [0, 1] such that when restricted on G/H × {0} and G/H × {1}, g coincides withĝ 1 andĝ 2 respectivel...

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On uniqueness and existence of conformally compact Einstein metrics with homogeneous conformal infinity

2022

Calc. Var.

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Rigidity of conformally compact manifolds with the round sphere as the conformal infinity

Cited by 21 publications

References 10 publications

Regularity and rigidity of asymptotically hyperbolic manifolds

Regularity and rigidity of asymptotically hyperbolic manifolds

On uniqueness of conformally compact Einstein metrics with homogeneous conformal infinity

On uniqueness and existence of conformally compact Einstein metrics with homogeneous conformal infinity

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