Rigidity of Einstein manifolds with positive scalar curvature

Xu, Hongwei; Gu, Juan-Ru

doi:10.1007/s00208-013-0957-7

Cited by 15 publications

(5 citation statements)

References 26 publications

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“…Recently the authors [46] proved the following optimal rigidity theorem for Einstein manifolds, which provides an evidence for Yau Conjectures I and II. for n = 4k, k ∈ Z + [2, ∞) and k = 4, 3 5 for n = 16, then M is either diffeomorphic to S n , or isometric to a compact rank one symmetric space.…”

Section: )mentioning

confidence: 96%

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The sphere theorems for manifolds with positive scalar curvature

Gu¹,

Xu²

2012

J. Differential Geom.

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Some new differentiable sphere theorems are obtained via the Ricci flow and stable currents. We prove that if M n is a compact manifold whose normalized scalar curvature and sectional curvature satisfy the pointwise pinching condition R 0 > σ n K max , where σ n ∈ ( 1 4 , 1) is an explicit positive constant, then M is diffeomorphic to a spherical space form. This gives a partial answer to Yau's conjecture on pinching theorem. Moreover, we prove that if M n (n ≥ 3) is a compact manifold whose (n − 2)-th Ricci curvature and normalized scalar curvature satisfy the pointwise condition Ric (n−2) min > τ n (n−2)R 0 , where τ n ∈ ( 1 4 , 1) is an explicit positive constant, then M is diffeomorphic to a spherical space form. We then extend the sphere theorems above to submanifolds in a Riemannian manifold. Finally we give a classification of submanifolds with weakly pinched curvatures, which improves the differentiable pinching theorems due to Andrews, Baker and the authors. * 2010 Mathematics Subject Classification. 53C20; 53C40. exists for all time and converges to a constant curvature metric as t → ∞. Here r g(t) denotes the mean value of the scalar curvature of g(t).Theorem B([10]). Let (M, g 0 ) be a compact, locally irreducible Riemannian manifold of dimension n(≥ 4). Assume that M × R 2 has nonnegative isotropic curvature, i.e.,for all orthonormal four-frames {e 1 , e 2 , e 3 , e 4 } and all λ, µ ∈ [−1, 1]. Then one of the following statements holds: (i) M is diffeomorphic to a spherical space form.(ii) n = 2m and the universal cover of M is a Kähler manifold biholomorphic to CP m . (iii) The universal cover of M is isometric to a compact symmetric space.On the other hand, some important work on sphere theorems for manifolds with positive Ricci curvature have been made by several geometers (see [3, 14, 21, 30, 36, 39], etc.). In 1990's, Cheeger, Colding and Petersen [14, 30] proved the following differentiable sphere theorem for manifolds with positive Ricci curvature.Theorem C. Let M n be a compact and simply connected Riemannian n-manifold with Ricci curvature Ric M ≥ n − 1. Suppose that one of the following conditions holds:where ω n = vol(S n ) and ε 1 (n) is some positive constant; (ii) λ n+1 < n + ε 2 (n), where λ n+1 is the (n + 1)-th eigenvalue of M and ε 2 (n) is some positive constant. Then M is diffeomorphic to S n . Let K(π) be the sectional curvature of M for 2-plane π ⊂ T x M , Ric(u) the Ricci curvature of M for unit vector u ∈ U x M . Set K max (x) := max π⊂TxM K(π), Ric min (x) := min u∈UxM Ric(u). Inspired by Shen's topological sphere theorem [36], the authors [44] obtained the following differentiable sphere theorem for manifolds of positive Ricci curvatures.Theorem D. Let M n be a compact Riemannian n-manifold. If Ric min > δ n (n − 1)K max , where δ n = 1 − 6 5(n−1) , then M is diffeomorphic to a spherical space form. In particular, if M is simply connected, then M is diffeomorphic to S n . Let M n be a submanifold in a Riemannian manifold M N . Denote by H and S the mean curvature and the s...

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Section: )mentioning

confidence: 96%

“…Recently the authors [46] proved the following optimal rigidity theorem for Einstein manifolds, which provides an evidence for Yau Conjectures I and II. Theorem 3.4.…”

Section: Manifolds Of Positive Scalar Curvaturementioning

confidence: 96%

The sphere theorems for manifolds with positive scalar curvature

Gu¹,

Xu²

2012

J. Differential Geom.

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“…Because sec Mi ≥ 1 − 3 n+2 , we have sec M∞ ≥ 1 − 3 n+2 . Therefore it follows from Theorem 1.1 (ii) of [43] that M ∞ is isometric to a locally symmetric manifold finitely covered by S n or CP m .…”

Section: Applicationsmentioning

confidence: 93%

“…Remark 4.4. If we assume, in the situation of Corollary 4.3, that sec M ≥ 1− 3 n+2 +δ, with δ > 0, then by Corollary 3.1 of [43] M will be diffeomorphic to a spherical space form. Of course, in this case ǫ will also depend on δ.…”

Section: Applicationsmentioning

confidence: 99%

Smooth stability and sphere theorems for manifolds and Einstein manifolds with positive scalar curvature

Tuschmann

Wiemeler

2019

Communications in Analysis and Geometry

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“…[, Theorem 4.4]). See also for another sufficient conditions for an Einstein manifold to be locally symmetric.…”

Section: L‐parabolicity Of Linear Weingarten Hypersurfacesmentioning

confidence: 99%

Rigidity of linear Weingarten hypersurfaces in locally symmetric manifolds

et al. 2016

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Our purpose in this paper is to study the rigidity of complete linear Weingarten hypersurfaces immersed in a locally symmetric manifold obeying some standard curvature conditions (in particular, in a Riemannian space with constant sectional curvature). Under appropriated constrains on the scalar curvature function, we prove that such a hypersurface must be either totally umbilical or isometric to an isoparametric hypersurface with two distinct principal curvatures, one of them being simple. Furthermore, we also deal with the parabolicity of these hypersurfaces with respect to a suitable Cheng–Yau modified operator.

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Rigidity of Einstein manifolds with positive scalar curvature

Cited by 15 publications

References 26 publications

The sphere theorems for manifolds with positive scalar curvature

The sphere theorems for manifolds with positive scalar curvature

Smooth stability and sphere theorems for manifolds and Einstein manifolds with positive scalar curvature

Rigidity of linear Weingarten hypersurfaces in locally symmetric manifolds

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