The Yamabe problem

Lee, John M.; Parker, Thomas H.

doi:10.1090/s0273-0979-1987-15514-5

Cited by 977 publications

(908 citation statements)

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“…Here we show that any complete Riemannian manifold that already satisfies all our assumptions and the conclusion of Huber's theorem is necessarily conformally quasi-isometric to an asymptotically locally euclidean (ALE) manifold. This remark provides some information on the geometry near the punctures of the metrics studied in section 1: they are necessarily obtained by a stereographic conformal blow-up from the "compact" metric -in the terminology of [13,22]. It also aims at justifying our strategy in the next section, which reduces Huber's problem to finding ALE structures in the conformal class of the original metric.…”

Section: R(x) ρ} ρ Dρ < ∞mentioning

confidence: 99%

“…In fact, we shall prove the following, slightly stronger, result, which also gives some information on the case treated in section 2. We recall that an end E of a complete Riemannian manifold (M n , g) is said to be asymptotically euclidean (of order τ > 0) if E is diffeomorphic to the complement of a euclidean ball in the euclidean space R n and if, in theses coordinates, the metric g satisfies Keeping the terminology of [22], we shall say that Ω is obtained from (M, g 0 ) by a stereographic conformal blow up.…”

Section: On the Geometry Of Compact Manifolds With A Finite Number Ofmentioning

confidence: 99%

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The Huber theorem for non-compact conformally flat manifolds

Carron¹,

Herzlich²

2002

Commentarii Mathematici Helvetici

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Abstract.It was proved in 1957 by Huber that any complete surface with integrable Gauss curvature is conformally equivalent to a compact surface with a finite number of points removed. Counterexamples show that the curvature assumption must necessarily be strengthened in order to get an analogous conclusion in higher dimensions. We show in this paper that any non compact Riemannian manifold with finite L n/2 -norm of the Ricci curvature satisfies Huber-type conclusions if either it is a conformal domain with volume growth controlled from above in a compact Riemannian manifold or if it is conformally flat of dimension 4 and a natural Sobolev inequality together with a mild scalar curvature decay assumption hold. We also get partial results in other dimensions. Mathematics Subject Classification (2000). 53C21, 58J60.

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Section: R(x) ρ} ρ Dρ < ∞mentioning

confidence: 99%

Section: On the Geometry Of Compact Manifolds With A Finite Number Ofmentioning

confidence: 99%

The Huber theorem for non-compact conformally flat manifolds

Carron¹,

Herzlich²

2002

Commentarii Mathematici Helvetici

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“…The geometric meaning of this functional is that if u is a minimizer, or indeed any critical point, then the conformally related metric g = u 4 n−2 g has constant scalar curvature on any open set where u > 0. We refer to the well-known survey paper by Lee and Parker [16], as well as [27], [28], for all details on the complete existence theory in the setting of smooth compact manifolds.…”

Section: Introductionmentioning

confidence: 99%

The Yamabe problem on stratified spaces

2014

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We introduce new invariants of a Riemannian singular space, the local Yamabe and Sobolev constants, and then go on to prove a general version of the Yamabe theorem under that the global Yamabe invariant of the space is strictly less than one or the other of these local invariants. This rests on a small number of structural assumptions about the space and of the behavior of the scalar curvature function on its smooth locus. The second half of this paper shows how this result applies in the category of smoothly stratified pseudomanifolds, and we also prove sharp regularity for the solutions on these spaces. This sharpens and generalizes the results of Akutagawa and Botvinnik [3] on the Yamabe problem on spaces with isolated conic singularities.

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“…We finally note that the requirement R > 0, which ensures the solvability of the Lichnerowicz equation, could be reformulated in terms of a condition on the Yamabe number (cf. [29]). …”

Section: Introductionmentioning

confidence: 99%