The Huber theorem for non-compact conformally flat manifolds

Carron, Gilles; Herzlich, Marc

doi:10.1007/s00014-002-8336-0

Cited by 29 publications

(26 citation statements)

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“…Namely, the theory of ALE ends of conformally flat, half-conformally or, more generally, Bach-flat 4-manifolds, see [7], and e.g. [8], [31], [32]. In fact, we have the following result by S. Bando, A. Kasue and H. Nakajima; see Section 4 in [7].…”

Section: Proof Of the Abstract Resultsmentioning

confidence: 89%

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REMARKS ON L^p-VANISHING RESULTS IN GEOMETRIC ANALYSIS

Pigola

Veronelli

2012

Int. J. Math.

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Abstract. We survey some L p -vanishing results for solutions of Bochner or Simons type equations with refined Kato inequalities, under spectral assumptions on the relevant Schrödinger operators. New aspects are included in the picture. In particular, an abstract version of a structure theorem for stable minimal hypersurfaces of finite total curvature is observed. Further geometric applications are discussed. Introduction and some vanishing resultsThis paper originates from an attempt to understand the abstract content of a structure theorem for stable minimal hypersurfaces of finite total scalar curvature.(ii) The "stability operator" L = −∆ − |II| 2 has non-negative spectrum.Then, |II| ≡ 0, that is, f (M ) is an affine m-plane .

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Section: Proof Of the Abstract Resultsmentioning

confidence: 89%

“…It follows that the uniform decay estimate required on a (x) is a consequence of the assumption a ∈ L m/2 ; see e.g. [8], [25], [34] and references therein.…”

Section: Statement Of the New Abstract Resultsmentioning

confidence: 99%

REMARKS ON L^p-VANISHING RESULTS IN GEOMETRIC ANALYSIS

Pigola

Veronelli

2012

Int. J. Math.

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“…One such extension is also contained in the thesis of H. Fang ([44]). For conformal structures which are not necessarily locally conformally flat, there is an extension of Theorem 5.1 by G. Carron and M. Herzlich ( [19]). …”

Section: Sun-yung Alice Changmentioning

confidence: 93%

Conformal invariants and partial differential equations

Chang

2005

Bull. Amer. Math. Soc.

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“…In particular M has a finite number of ends. As it is proved in [4] that sup S(r) |K g | = o(r −2 ) in our setting, one can apply Tian-Viaclovsky's theorem if one has finiteness of the first Betti number.…”

mentioning

confidence: 92%

“…In [4], we consider a complete conformally flat Riemannian manifold (M n , g) which satisfies the Sobolev inequality :…”

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confidence: 99%

Erratum to “The Huber theorem for non-compact conformally flat manifolds”

Carron¹,

Herzlich²

2007

Comment. Math. Helv.

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Abstract. An argument in our paper The Huber theorem for non-compact conformally flat manifolds [Comment. Math. Helv. 77 (2002), 192-220] was not justified. Using recent work by G. Tian and J.Viaclovsky, we show that our result holds true.In [4], we consider a complete conformally flat Riemannian manifold (M n , g) which satisfies the Sobolev inequality :and whose Ricci tensor is in L n 2 . On page 208 we then assert that "diameter is controlled from above and volume growth is controlled from below (from the Sobolev inequality) on each annulus M kr −M k −1 r . We can then infer from Anderson-Cheeger harmonic radius' theory that the rescaled annuli (by a finite (and uniformly bounded ) number of balls of uniformly bounded size where the metric coefficients are C 1,α -close to the euclidean metric." In fact, the trivial extrinsic diameter bound is not enough to ensure Anderson-Cheeger compactness (one needs an intrinsic diameter control) and this argument needs to be justified. This is what we intend to do below.First observe that the nedeed intrisic diameter control can be replaced by an upper bound on the volume growth of geodesic balls. Recently, G. Tian and J. Viaclovsky [5] investigated an issue closely related to ours, and proved the following result:) be a complete noncompact Riemannian manifold of dimension n ≥ 3. If there exists a constant C 1 > 0 such that vol(B(q, s)) ≥ C 1 s n , for any q ∈ X, s ≥ 0, that furthermore sup S(r) |K g | = o(r −2 ) as r → ∞, where S(r) is the sphere of radius r centered at a basepoint p, and that b 1 (X) < ∞, then there exists a constant C 2 so thatUsing this one can show that our argument remains true. Indeed, the Sobolev inequality implies that the manifold has an euclidean lower bound on the volume growth of geodesic balls. Moreover, Sobolev and L n 2 -integrability of the Ricci curvature imply that the space of L 2 harmonic 1-formsand the first cohomology group with compact support H 1 c (M ) have finite dimensions [2,3]. In particular M has a finite number of ends. As it is proved in [4] that 1

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The Huber theorem for non-compact conformally flat manifolds

Cited by 29 publications

References 32 publications

REMARKS ON L^p-VANISHING RESULTS IN GEOMETRIC ANALYSIS

REMARKS ON L^p-VANISHING RESULTS IN GEOMETRIC ANALYSIS

Conformal invariants and partial differential equations

Erratum to “The Huber theorem for non-compact conformally flat manifolds”

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The Huber theorem for non-compact conformally flat manifolds

Cited by 29 publications

References 32 publications

REMARKS ON Lp-VANISHING RESULTS IN GEOMETRIC ANALYSIS

REMARKS ON Lp-VANISHING RESULTS IN GEOMETRIC ANALYSIS

Conformal invariants and partial differential equations

Erratum to “The Huber theorem for non-compact conformally flat manifolds”

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REMARKS ON L^p-VANISHING RESULTS IN GEOMETRIC ANALYSIS

REMARKS ON L^p-VANISHING RESULTS IN GEOMETRIC ANALYSIS