Abstract: The main purpose of our paper is to understand the structure of conformally compact manifolds by studying the space of L² harmonic 1-forms on it. First following the Wang’s method and using the condition for curvature and the first eigenvalue, we know that either there does not exist any nontrivial L² harmonic 1-form or some differential equations hold on these conformally compact manifolds. By solving these equations, we can prove that the given manifolds can be written as the product of a Euclidean space and a totally geodesic submanifolds whose curvature has lower bound, and the metrics of the manifolds can be expressed explicitly. For the generalized complete manifolds, if we restrict the growth of the L² harmonic 1-forms on them, the similar theorem holds, too.

Key words: conformally compact manifold, Ricci curvature, the first eignvalue, L² harmonic 1-forms