A rigidity theorem for complete noncompact Bach-flat manifolds

Chu, Yawei

doi:10.1016/j.geomphys.2010.11.003

Cited by 8 publications

(8 citation statements)

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“…For an n-dimensional Riemannian manifold with constant scalar curvature, the Laplacian of the norm square of trace-free curvature tensor was obtained in [4].…”

Section: Remarkmentioning

confidence: 99%

Complete noncompact manifolds with harmonic curvature

Chu

2012

Front. Math. China

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Let (M n , g) be an n-dimensional complete noncompact Riemannian manifold with harmonic curvature and positive Sobolev constant. In this paper, by employing an elliptic estimation method, we show that (M n , g) is a space form if it has sufficiently small L n/2 -norms of trace-free curvature tensor and nonnegative scalar curvature. Moreover, we get a gap theorem for (M n , g) with positive scalar curvature.

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“…For an n-dimensional Riemannian manifold with constant scalar curvature, the Laplacian of the norm square of trace-free curvature tensor was obtained in [4].…”

Section: Remarkmentioning

confidence: 99%

Complete noncompact manifolds with harmonic curvature

Chu

2012

Front. Math. China

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“…Later, the first author [12] improved Kim's result and showed that, under the same assumptions in [11], M 4 is in fact a space of constant curvature. …”

Section: Introductionmentioning

confidence: 95%

“…For example, any simply connected complete locally conformally flat manifold has the positive Yamabe constant (see [16]). In order to prove Theorem 1.1, we need the following formula proved in [12]. …”

Section: Introductionmentioning

confidence: 99%

Rigidity of complete noncompact bach-flat n-manifolds

Chu

Pinghua

2012

Journal of Geometry and Physics

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a b s t r a c t Let (M n , g) be a complete noncompact Bach-flat n-manifold with the positive Yamabe constant and constant scalar curvature. Assume that the L 2 -norm of the trace-free Riemannian curvature tensor • R m is finite. In this paper, we prove that (M n , g) is a constant curvature space if the L n 2 -norm of • R m is sufficiently small. Moreover, we get a gap theorem for (M n , g) with positive scalar curvature. This can be viewed as a generalization of our earlier results of 4-dimensional Bach-flat manifolds with constant scalar curvature R ≥ 0 [Y.W. Chu, A rigidity theorem for complete noncompact Bach-flat manifolds, J. Geom. Phys. 61 (2011) 516-521]. Furthermore, when n > 9, we derive a rigidity result for R < 0.

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“…There are many other ✏-rigidity results that rely on a priori functional inequalities (such as a Sobolev inequality or as the above Hardy inequality) and integral bounds on the curvature (cf. for instance [5,13,20,21,25,27,29,30,32,33], [35, Theorem 7.1], [38]). Such results have been shown recently for critical metrics by G. Tian and J. Viaclovsky in dimension 4 and by X-X.…”

Section: Introductionmentioning

confidence: 99%