The total Q-curvature, volume entropy and polynomial growth polyharmonic functions

In this paper, we investigate a conformally flat and complete manifold $(M,g)=(\mathbb{R}^n,e^{2u}|dx|^2)$ with finite total Q-curvature. We introduce a new volume entropy, incorporating the background Euclidean metric, and demonstrate that the metric $g$ is normal if and only if the volume entropy is finite. Furthermore, we establish an identity for the volume entropy utilizing the integrated Q-curvature. Additionally, under normal metric assumption, we get a result concering the behavior of the geometric distance at infinity compared with Euclidean distance. With help of this result, we prove that each polynomial growth polyharmonic function on such manifolds is of finite dimension. Meanwhile, we prove several rigidity results by imposing restrictions on the sign of the Q-curvature. Specifically, we establish that on such manifolds, the Cohn-Vossen inequality achieves equality if and only if each polynomial growth polyharmonic function is a constant.