Yamabe Flow and Myers Type Theorem on Complete Manifolds

Abstract

Abstract. In this paper, we prove the following Myers-type theorem: if (M n , g), n ≥ 3, is an n-dimensional complete locally conformally flat Riemannian manifold with bounded Ricci curvature satisfying the Ricci pinching condition Rc ≥ ǫRg > 0, where ǫ > 0 is an uniform constant, then M n must be compact. Mathematics Subject Classification (2000): 35J60, 53C21, 58J05

Application of the Generalized Bochner Technique to the Study of Conformally Flat Riemannian Manifolds

“…For this, the methods of geometric analysis were developed (e.g., [3][4][5]). As a result, vanishing theorems of the classical Bochner technique took the form of Liouville type theorems, e.g., [5][6][7]. In this article, we discuss the global aspects of the geometry of locally conformally flat complete and compact Riemannian manifolds using the generalized and classical Bochner technique, as well as the Ricci flow, e.g., [8,9].…”

“…Articles [6,7] are devoted to the search for analytic compactness conditions for complete locally conformally flat Riemannian manifolds. The main result of [6] is that an n-dimensional (n ≥ 3) simply connected complete locally conformally flat Riemannian manifold with positive constant scalar curvature is compact if M Ric p d vol g < ∞ for all p ≥ n/2, where Ric = Ric −(s/n) g is the traceless Ricci tensor.…”

“…Remark 3. According to [7], a complete locally conformally flat Riemannian manifold (M, g) of dimension n ≥ 3 is compact if its Ricci tensor is bounded and satisfies the inequality Ric ≥ ε s g for some constant ε > 0, where s is the scalar curvature. Recall (see [20]) that in an orthonormal basis {e 1 , .…”

Properties of complete non-compact Yamabe solitons

“…This decay property may be useful to understand the geometry of the non-locally conformally flat complete non-compact Riemannian manifolds. In the previous work [12] we have proved that locally conformally flat complete riemannian manifolds are compact. …”

Yamabe flow and ADM mass on asymptotically flat manifolds

“…So we omit the details here, and one may see Lemma 6.19 and Lemma 6.20 in [13] for the proof. for all x ∈ B g 0 (p, r 2 ) and t ∈ (0, τ ] (see [27]). Finally, we give the proofs of Theorem 1.4 and Theorem 1.5.…”