Complete noncompact manifolds with harmonic curvature

Abstract

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.

On Compact Manifolds with Harmonic Curvature and Positive Scalar Curvature

“…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.…”

Some L p Rigidity Results for Complete Manifolds with Harmonic Curvature

“…Remark 2.2. Although Lemma 2.1 has been proved in [9], we give an explicit coefficient of the term | Rm| 3 in ( 8). When M is a complete locally conformally flat Riemannian n-manifold, it follows from (10) that…”

“…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.…”