Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{p}$\end{document} harmonic 1-forms on conformally flat Riemannian manifolds

In this paper, we establish a finiteness theorem for Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{p}$\end{document} harmonic 1-forms on a locally conformally flat Riemannian manifold under the assumptions on the Schrödinger operators involving the squared norm of the traceless Ricci form. This result can be regarded as a generalization of Han’s result on L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{2}$\end{document} harmonic 1-forms.


Introduction
Investigating the relationship between the geometry and topology of a Riemannian manifold M and the spaces of harmonic forms is one of the most important problems in differential geometry. Thanks to the Hodge theory, it is known that when M is compact, the space of harmonic 1-forms on M is isomorphic to its first de Rham cohomology group. When M is noncompact, the Hodge theory is no longer applicable, and it is natural to consider L 2 harmonic forms. Furthermore, L 2 Hodge theory holds for complete noncompact manifolds (see, e.g., [1,9]), just like classical Hodge theory works on the compact case. Particularly, Li and Tam [21] showed that the theory of L 2 harmonic 1-forms can be used for understanding the topology at infinity of a complete Riemannian manifold.
Recall that a Riemannian manifold (M m , g) of dimension m is said to be locally conformally flat if it admits a coordinate covering (U α , ϕ α ) such that the map ϕ α : (U α , g α ) → (S m , g 0 ) is a conformal map, where g 0 is the standard metric on S m . A conformally flat Riemannian manifold may be regarded as a generalization of a Riemannian surface because every two-dimensional Riemannian manifold is locally conformally flat. However, not all higher-dimensional manifolds have locally conformally flat structure, and giving classification of locally conformally flat manifolds is important as well as difficult. However, under various geometric conditions, there are substantial research results on the classification of conformally flat Riemannian manifolds (see [2,5,6,14,18,22,29] for details).
For L 2 harmonic forms, Lin [23] proved some vanishing and finiteness theorems for L 2 harmonic 1-forms on a locally conformally flat Riemannian manifold that satisfies an integral pinching condition on the traceless Ricci tensor and for which the scalar curvature is nonpositive or satisfies some integral pinching conditions. Dong et al. [10] proved van-ishing theorems for L 2 harmonic p-forms on a complete noncompact locally conformally flat Riemannian manifold under suitable conditions. Similarly, Han [16] obtained some vanishing and finiteness theorems for L 2 harmonic 1-forms on a locally conformally flat Riemannian manifold under the assumptions on the Schrödinger operators involving the squared norm of the traceless Ricci form. Moreover, many results showed that there is a close correlation between the topologies of the submanifolds and L 2 harmonic 1-forms; see [3,11,12,24,30,31] and the references therein.
The results of L 2 harmonic forms make L 2 theory on manifolds clearer and easier to understand as compared to general L p theory (see [26]). For L p harmonic 1-forms, Han et al. [18] obtained some vanishing and finiteness theorems for L p p-harmonic 1-forms on a locally conformally flat Riemannian manifold with some assumptions. Analogously, there is substantial research indicating that the topologies of the submanifolds is closely associated with L p harmonic 1-forms; see [4,7,8,15,17,19,22] and the references therein.
Meanwhile, Lin [24] studied the relations between the index of the Schrödinger operator L = + m-1 m |B| 2 and the topology of M m , where B is the second fundamental form of M m , and M m is a complete noncompact minimal submanifold of dimension m immersed in R m+n . In particular, when M m is an m-dimensional locally conformally flat Riemannian manifold, Han [16] focused on the Schrödinger operator L = + |T| and investigated the relations between it and the topological structure of M m , where Ric, R, and T = Ric-R m g are the Ricci curvature tensor, the scalar curvature, and the traceless Ricci tensor of (M m , g), respectively.
Inspired by Han's work [16] and the above mentioned aspects, in this paper, we investigate relations between the index of a Schrödinger operator L = + |T| of the locally conformally flat manifold M m and the space of L p harmonic 1-forms on M m . We prove that if the index of a Schrödinger operator is finite and M |R| m 2 dv < ∞, then for certain p > 0, the dimension of H 1 (L 2p (M)) is finite. This can be regarded as a generalization of Han's results [16] for the space of L 2 harmonic 1-forms.
In this paper, we obtain the following finiteness theorem for the space of L p harmonic 1-forms.
where H 1 (L 2p (M)) denotes the space of L 2p harmonic 1-forms on M.

Preliminaries
Consider an elliptic operator L = +Q on M m , whereQ is the smooth potential of it. Let D be a relatively compact domain of M m , and let ind(L D ) be the number of negative eigenvalues of L with Dirichlet boundary condition: Let (M m , g) be complete locally flat Riemannian manifold of dimension m, and let be the Hodge Laplace-Beltrami operator of M m that acts on the space of differentialp-forms. From the Weitzenböck formula [28] we know that where ∇ 2 is the Bochner Laplacian, and Kp is an endomorphism depending on the curvature of M m . By choosing an orthonormal basis {θ 1 , . . . , θ m } dual to {e 1 , . . . , e m }, we can express Kp as forp-forms ω. In particular, when ω is a 1-form and ω expresses the vector field dual to ω, we have We also need the following lemmas, which are important tools in proving our result. , where ω m is the volume of the unit sphere in R m . Hence we have the inequality for any f ∈ C ∞ 0 (M). By using (1) Lin [23] obtained the following result.
for some constant C(m) > 0, which is equal to Q(S m ) -1 in the case of R ≤ 0, where f ∈ C ∞ 0 (M). In particular, M has infinite volume.
for any x ∈ M and r > 0.
On the other hand, applying the Sobolev inequality (2) to the term η|ω| p , we have where C(m) > 0 is the Sobolev constant. From (16) and (17) we have where C 1 (m, p) = C(m) (1 + D(m, p)) is a positive constant depending only on m and p. From here the proof mainly follows by the standard techniques (e.g., see [3]). Applying the definition of η to inequality (18), we get D(m, p)) is a positive constant depending only on m and p. Then, letting r → ∞ and noting that |ω| ∈ L 2p (M), we have By Hölder's inequality we conclude that From (19) and (20) we have that is, where C 3 is a positive constant depending only on Vol(B x 0 (r 0 + 2)), m, and p.
This, together with (32), yields that dim W is upper bounded by a fixed constant, that is, dim W ≤ C 7 , where C 7 depends only on m, p, Vol(B x 0 (r 0 + 2)), and sup B x 0 (r 0 +2) F. Then we obtain that dim H 1 (L 2p (M)) < ∞. This completes the proof of Theorem 1.1.

Conclusions
In this paper, we study the dimension of the space of L p harmonic 1-forms on a locally conformally flat Riemannian manifold M m . The key to our research is Lemma 2.3 for L 2p harmonicq-forms [20,25] and the geometric analysis techniques. With their help, we prove that the dimension of the space of L p harmonic 1-forms must be finite for certain p under the assumptions on the Schrödinger operators involving the squared norm of the traceless Ricci form. This result can be regarded as a generalization of Han's result [16] for L 2 harmonic 1-forms.