  In this paper we introduce a new geometric flow --- the hyperbolic gradient
flow for graphs in the $(n+1)$-dimensional Euclidean space $\mathbb{R}^{n+1}$.
This kind of flow is new and very natural to understand the geometry of
manifolds. We particularly investigate the global existence of the evolution of
convex hypersurfaces in $\mathbb{R}^{n+1}$ and the evolution of plane curves,
and prove that, under the hyperbolic gradient flow, they converge to the
hyperplane and the straight line, respectively, when $t$ goes to the infinity.
Our results show that the theory of shock waves of hyperbolic conservation laws
can be naturally applied to do surgery on manifolds. Some fundamental but open
problems are also given.
