A sharp Sobolev inequality on Riemannian manifolds

Li, Yan Yan; Ricciardi, Tonia

doi:10.1016/s1631-073x(02)02529-3

Cited by 13 publications

(15 citation statements)

References 45 publications

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“…Comparing with some existing results for higher dimensional Yamabe type problem (see, for example, [18] and [20]), our proof seems quite standard. We start with a local sharp inequality (caution: this local inequality does not rely on Moser's inequality), which allows us to obtain a rough sharp inequality (or, sometimes, we refer it as an ǫ-level inequality) in Proposition 2.1.…”

Section: 3supporting

confidence: 73%

Liouville Energy on a Topological Two Sphere

Chen

Zhu

2013

Commun. Math. Stat.

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In this paper we shall give an analytic proof of the fact that the Liouville energy on a topological two sphere is bounded from below. Our proof does not rely on the uniformization theorem and the Onofri inequality, thus it is essentially needed in the alternative proof of the uniformization theorem via the Calabi flow. Such an analytic approach also sheds light on how to obtain the boundedness for E 1 energy in the study of general Kähler manifolds.1 In a recent note, Chen, Lu and Tian [11] clarify that the Ricci flow did yield another proof of the uniformization theorem for spheres even without using current integral estimates. Nevertheless our result not only answer a conjecture of Chen [8], but also has broad applications in the study of Kähler manifolds (see the historic note).

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Section: 3supporting

confidence: 73%

Liouville Energy on a Topological Two Sphere

Chen

Zhu

2013

Commun. Math. Stat.

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“…For manifolds with boundary, the same type of result (see Li and Zhu [14]) is true. More precisely, if (V , g) is a compact Riemannian n-manifold with smooth boundary and n ≥ 3, then, for all u ∈ C ∞ (V ),…”

Section: Introductionmentioning

confidence: 55%

A Poincaré-Sobolev type inequality on compact Riemannian manifolds with boundary

Holcman

Humbert²

2001

Math Z

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In this paper we study a Poincaré-Sobolev type inequality on compact Riemannian n-manifolds with boundary where the exponent growth is critical. Two constants have to be determined. We show that, contrary to the classical Sobolev inequality, the first best constant in this inequality does not depend on the dimension only, but depends on the geometry. It can be represented as the minimum µ of a given energy functional. We study the nonlinear PDE associated to this functional which involves the geometry of the boundary. For a star-shaped domain D in R n whose boundary has positive Ricci curvature, we give explicitly two Sobolev constants corresponding to the embedding H 1 (D) in L 2(n−1) n−2 (∂D). This result is used to obtain an explicit geometrical lower bound for µ.

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“…However, several difficulties appear because we have to study functions on M while we have informations only on ∂M . The specific problems of trace inequalities can also be seen in the paper of Li and Zhu [12] who obtained the same result for trace Sobolev inequality. Their proof is really different from the one of Hebey and Vaugon [8] who solved the problem for standard Sobolev inequality (without trace).…”

Section: Introductionmentioning

confidence: 63%

“…One easily gets that, in addition, limL k = limλ 2 k = 1. Coming back to (12), it follows that, if (c α ) α is such that lim Aα cα = 0,…”

Section: Emmanuel Humbert Nodeamentioning

confidence: 99%