Constantes explicites pour les inégalités de Sobolev sur les variétés riemanniennes compactes

“…Explicit upper bounds for S can be given in special geometries, like, see Ilias [13], when the Ricci curvature of the manifold is positive. Concerning lower bounds, it is well-known that S ≥ K 2 ⋆ n , where K n is the sharp Sobolev constant in the n-dimensional Euclidean space for the Sobolev inequality u L 2 ⋆ ≤ K n ∇u L 2 .…”

Section: )mentioning

confidence: 99%

A Variational Analysis of Einstein–Scalar Field Lichnerowicz Equations on Compact Riemannian Manifolds

Hebey

Pacard

Pollack

2007

Commun. Math. Phys.

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Abstract. We establish new existence and non-existence results for positive solutions of the Einstein-scalar field Lichnerowicz equation on compact manifolds. This equation arises from the Hamiltonian constraint equation for the Einstein-scalar field system in general relativity. Our analysis introduces variational techniques, in the form of the mountain pass lemma, to the analysis of the Hamiltonian constraint equation, which has been previously studied by other methods.

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“…But thanks to Theorem 2.1 we already know that the equality is attained for ϕ P , and therefore we have equality in each line of (12). This implies that dv 0 vanishes and v 0 is constant on each connected component of U + , and since U + is connected, the quotient v 0 = −ϕ P /ϕ N is constant on U + .…”

Section: Obata Singular Theoremmentioning

confidence: 88%

“…A Sobolev inequality of the previous form was proven by S. Ilias in [12] for compact smooth manifolds with Ricci tensor bounded by below by a positive constant, and by D. Bakry in [4] for a much general setting. Our proof is inspired by the argument due to D. Bakry.…”

Section: Preliminariesmentioning

confidence: 97%

An Obata singular theorem for stratified spaces

Mondello

2017

Trans. Amer. Math. Soc.

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Abstract. Consider a stratified space with a positive Ricci lower bound on the regular set and no cone angle larger than 2π. For such stratified space we know that the first non-zero eigenvalue of the Laplacian is larger than or equal to the dimension. We prove here an Obata rigidity result when the equality is attained: the lower bound of the spectrum is attained if and only if the stratified space is isometric to a spherical suspension. Moreover, we show that the diameter is at most equal to π, and it is equivalent for the diameter to be equal to π and for the first non-zero eigenvalue of the Laplacian to be equal to the dimension. We finally give a consequence of these results related to the Yamabe problem. Consider an Einstein stratified space without cone angles larger than 2π: if there is a metric conformal to the Einstein metric and with constant scalar curvature, then it is an Einstein metric as well. Furthermore, if its conformal factor is not a constant, then the space is isometric to a spherical suspension. MSC2010: 53 Differential geometry, 58 Global analysis, analysis on manifolds.

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Constantes explicites pour les inégalités de Sobolev sur les variétés riemanniennes compactes

Cited by 52 publications

References 5 publications

Isoperimetric profile comparisons and Yamabe constants

Isoperimetric profile comparisons and Yamabe constants

A Variational Analysis of Einstein–Scalar Field Lichnerowicz Equations on Compact Riemannian Manifolds

An Obata singular theorem for stratified spaces

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