Geometries and Topologies of Conformally Flat Riemannian Manifolds
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Geometries and Topologies of Conformally Flat Riemannian Manifolds Jing Li1 · Shuxiang Feng2 · Peibiao Zhao1 Received: 21 May 2019 / Accepted: 9 January 2020 © Iranian Mathematical Society 2020
Abstract The present paper investigates vanishing and finiteness theorems for L 2B harmonic 1forms on a locally conformally flat Riemannian manifold with a Schrödinger operator |R| . Furthermore, based on these vanishing and finiteness theorems and L = Δ+ √ n the theory of L 2 harmonic 1-forms by Li–Tam, this paper derives that the locally |R| conformally flat Riemannian manifold with a Schrödinger operator L = Δ + √ n has one-end and finite ends. The results posed here can be regarded as a natural generalization of the work by Han (Results Math 73:54, 2018). Keywords Harmonic 1-form · Conformally flat · Finite index · Nonparabolic end Mathematics Subject Classification 53C21 · 53C25
1 Introduction The study of geometries and topologies of conformally flat Riemannian manifolds has been an active field over the past 5 decades, and which is of great value to the classification of conformally flat Riemannian manifolds. There are substantial research
Communicated by Fatemeh Helen Ghane.
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Peibiao Zhao [email protected] Jing Li [email protected] Shuxiang Feng [email protected]
1
School of Science, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, People’s Republic of China
2
School of Mathematics and Statistics, Southwest University, Chongqing 400715, People’s Republic of China
123
Bulletin of the Iranian Mathematical Society
results on the classification of conformally flat Riemannian manifolds (one can see [1,4,5,9,27] for details). On the other hand, for the research on the geometry and topology of conformally flat Riemannian manifolds, there are also many celebrated works. For instance, Carron [2] showed H 1 (L 2 (M)) is isomorphic to the first-reduced L 2 -cohomology group of M, and derived that the dimension of H 1 (L 2 (M)) has an upper bound reaching the number of non-parabolic ends of M. Pigola et al. [23] obtained a vanishing theorem for bounded harmonic n-forms on a 2n-dimensional complete locally conformally flat manifold under the scalar curvature and volume growth conditions. Li and Wang [17] showed that for a complete, simply connected, locally conformally flat manifold M n (n ≥ 4) with scalar curvature R ≥ 0, if the Ricci curvature Ric ≥ 41 R and the scalar curvature satisfies some decay conditions, then either M has only one end, or M = R × N with a warped product metric for some compact manifold N . Lin [20] proved some vanishing and finiteness theorems for L 2 harmonic 1-forms on a locally conformally flat Riemannian manifold which satisfies some conditions of the traceless Ricci tensor and the scalar curvature. Similarly, Han [10] obtained some vanishing and finiteness theorems for L 2 harmonic 1-forms on a locally conformally flat Riemannian manifold under the condition of the Schrödinger operator involving the squared norm of the traceless Ricci form. Recently, Han et
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