Some old and new results about rigidity of critical metric

Carron, Gilles

doi:10.48550/arxiv.1012.0685

Cited by 3 publications

(4 citation statements)

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“…The another reason for this study on the metric with harmonic curvature is the fact that a Riemannian manifold has harmonic curvature if and only if the Riemannian connection is a solution of the Yang-Mills equations on the tangent bundle [4]. The complete manifolds with harmonic curvature have been studied in literature (e.g., [5,6,9,11,14,18,21,22,25,28,29,30]). Some isolation theorems of Weyl curvature tensor of positive Einstein manifolds are given in [7,15,16,18,21,28], when its L p -norm is small.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On Compact Manifolds with Harmonic Curvature and Positive Scalar Curvature

2017

J Geom Anal

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Abstract. Let M n (n ≥ 3) be an n-dimensional compact Riemannian manifold with harmonic curvature and positive scalar curvature. Assume that M n satisfies some integral pinching conditions. We give some rigidity theorems on compact manifolds with harmonic curvature and positive scalar curvature. In particular, Theorem 1.4, Corollary 1.6 and Theorem 1.9 are sharp for our conditions have the additional properties of being sharp. By this we mean that we can precisely characterize the case of equality.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

On Compact Manifolds with Harmonic Curvature and Positive Scalar Curvature

2017

J Geom Anal

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“…The another reason for this study on the metric with harmonic curvature is the fact that a Riemannian manifold has harmonic curvature if and only if the Riemannian connection is a solution of the Yang-Mills equations on the tangent bundle [4]. In recent years, the complete manifolds with harmonic curvature have been studied in literature (e.g., [5,9,14,15,18,20,21,22,26,27,28]). In particular, G. Tian and J. Viaclovsky [28], and X. Chen and B. Weber [8] have obtained ǫ-rigidity results for critical metric which relies on a Sobolev inequality and a integral bounds on the curvature in dimension 4 and in higher dimension, respectively.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Remark 1.5. Some L n 2 and L n trace-free Riemannian curvature pinching theorems have been shown by Kim [21], Chu [5], and Fu etc. [14], in which the constant C is not explicit, respectively.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Some L p Rigidity Results for Complete Manifolds with Harmonic Curvature

Xiao

2017

Potential Anal

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Let (M n , g)(n ≥ 3) be an n-dimensional complete Riemannian manifold with harmonic curvature and positive Yamabe constant. Denote by R andRm the scalar curvature and the trace-free Riemannian curvature tensor of M , respectively. The main result of this paper states thatRm goes to zero uniformly at infinity if for p ≥ n 2 , the L p -norm ofRm is finite. Moreover, If R is positive, then (M n , g) is compact. As applications, we prove that (M n , g) is isometric to a spherical space form if for p ≥ n 2 , R is positive and the L p -norm ofRm is pinched in [0, C 1 ), where C 1 is an explicit positive constant depending only on n, p, R and the Yamabe constant.In particular, we prove an L p ( n 2 ≤ p < n−2 2 (1+ 1 − 4 n ))-norm ofRic pinching theorem for complete, simply connected, locally conformally flat Riemannian n(n ≥ 6)-manifolds with constant negative scalar curvature.We give an isolation theorem of the trace-free Ricci curvature tensor of compact locally conformally flat Riemannian n-manifolds with constant positive scalar curvature, which improves Thereom 1.1 and Corollary 1 of E. Hebey and M. Vaugon [18]. This rsult is sharped, and we can precisely characterize the case of equality.

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“…Theorem 3.4 ([TV05a]. [TV08][Carron10]) Let (M, g) be a complete Riemannian manifold or Riemannian multi -fold with finite point singularities, g is critical metric (for example, Bach flat with zero scalar curvature). Assume that (M n , g) satisfies the Sobolev inequality,…”

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confidence: 99%