In this paper, we establish general monotone formulas to derive geometric inequalities related to k-admissible solution u in $R^n\setminus\Omega$, where $\Omega$ is smooth, k-convex and star-shaped domain, which constructed by Ma-Zhang and Xiao. As an application, we deal with an overdetermined problem for the k-Hessian equation ($1\leq k<\frac n2$) in the exterior domain by combining two integral identities and geometric inequalities (Ma-zhang's inequality and special Aleksandrov-Fenchel inequalities). As a consequence, we prove the corresponding ball characterizations which generalized Brandolini-Nitsch -Salani's results. This is joint work with Xingjian Zhou.
