On the asymptotic behavior of Einstein manifolds with an integral bound on the Weyl curvature

Gicquaud, Romain; Ji, Dandan; Shi, Yuguang

doi:10.4310/cag.2013.v21.n5.a8

Cited by 2 publications

(2 citation statements)

References 16 publications

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“…More generally, these special tensors can be considered on pinched Cartan-Hadamard Einstein manifolds (M n , , ), since Ric = R n , and by Schur's lemma the scalar curvature of Einstein manifolds of dimension n ≥ 3 must be constant. We highlight that complete noncompact Einstein manifolds with Ric = −(n − 1) , have a very special behavior at infinity, see Gicquaud et al [9]. Another example is obtained from Ŝ = S − tr(S) , on pinched Cartan-Hadamard manifolds (M n , , ), n ≥ 3, in this case, we have div Ŝ = 0, since divS = dtr(S).…”

Section: Preliminariesmentioning

confidence: 82%

Eigenvalue estimates of the drifted Cheng-Yau operator on bounded domains in pinched Cartan-Hadamard manifolds

M.¹,

Gomes²

2021

Preprint

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We show how a Bochner type formula can be used to establish universal inequalities for the eigenvalues of drifted Cheng-Yau operator on a bounded domain in a pinched Cartan-Hadamard manifold with the Dirichlet boundary condition. In the first theorem, the hyperbolic space case is treated in an independent way. For the more general setting, we first establish a Rauch comparison theorem for the Cheng-Yau operator and two estimates associated with the Bochner type formula for this operator. Next, we get some integral estimates of independent interest. As an application, we compute our universal inequalities. In particular, we obtain the corresponding inequalities for both Cheng-Yau operator and drifted Laplacian cases, and we recover the known inequalities for the Laplacian case. We also obtain a rigidity result for a Cheng-Yau operator on a class of bounded annular domains in a pinched Cartan-Hadamard manifold. In particular, we can use the potential function of the Gaussian shrinking soliton to obtain such rigidity for the Euclidean space case.

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Section: Preliminariesmentioning

confidence: 82%

Eigenvalue estimates of the drifted Cheng-Yau operator on bounded domains in pinched Cartan-Hadamard manifolds

M.¹,

Gomes²

2021

Preprint

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“…Note however that the proof given in this article is wrong because of a confusion between "closed geodesics" and "geodesic loop". The correct argument appears in [19]. We reproduce it here for the sake of completeness.…”

Section: Harmonic Coordinates Harmonic Radius and Applicationsmentioning

confidence: 86%

Conformal Compactification of Asymptotically Locally Hyperbolic Metrics II: Weakly ALH Metrics

Gicquaud¹

2013

Communications in Partial Differential Equations

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In this paper we pursue the work initiated in [6,7]: study the extent to which conformally compact asymptotically hyperbolic metrics can be characterized intrinsically. We show how the decay rate of the sectional curvature to −1 controls the Hölder regularity of the compactified metric. To this end, we construct harmonic coordinates that satisfy some Neumann-type condition at infinity. Combined with a new integration argument, this permits us to recover to a large extent our previous result without any decay assumption on the covariant derivatives of the Riemann tensor.

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On the asymptotic behavior of Einstein manifolds with an integral bound on the Weyl curvature

Cited by 2 publications

References 16 publications

Eigenvalue estimates of the drifted Cheng-Yau operator on bounded domains in pinched Cartan-Hadamard manifolds

Eigenvalue estimates of the drifted Cheng-Yau operator on bounded domains in pinched Cartan-Hadamard manifolds

Conformal Compactification of Asymptotically Locally Hyperbolic Metrics II: Weakly ALH Metrics

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