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In this note, we study the dynamics and associated zeta functions of conformally compact manifolds with variable negative sectional curvatures. We begin with a discussion of a larger class of manifolds known as convex co-compact manifolds with variable negative curvature. Applying results from dynamics on these spaces, we obtain optimal meromorphic extensions of weighted dynamical zeta functions and asymptotic counting estimates for the number of weighted closed geodesics. A meromorphic extension of the standard dynamical zeta function and the prime orbit theorem follow as corollaries. Finally, we investigate interactions between the dynamics and spectral theory of these spaces.
In this paper, we establish a compactness result for a class of conformally compact Einstein metrics defined on manifolds of dimension $dge 4$. As an application, we derive the global uniqueness of a class of conformally compact Einstein metric defin
We classify quadruples $(M,g,m,tau)$ in which $(M,g)$ is a compact Kahler manifold of complex dimension $m>2$ with a nonconstant function $tau$ on $M$ such that the conformally related metric $g/tau^2$, defined wherever $tau e 0$, is Einstein. It tur
We characterize manifolds which are locally conformally equivalent to either complex projective space or to its negative curvature dual in terms of their Weyl curvature tensor. As a byproduct of this investigation, we classify the conformally complex
We show that locally conformally flat quasi-Einstein manifolds are globally conformally equivalent to a space form or locally isometric to a $pp$-wave or a warped product.
We study the existence of closed geodesics on compact Riemannian orbifolds, and on noncompact Riemannian manifolds in the presence of a cocompact, isometric group action. We show that every noncontractible Riemannian manifold which admits such an act