Hyperbolic Gradient Flow: Evolution of Graphs in R^{n+1}

Details Hyperbolic Gradient Flow: Evolution of Graphs in R^{n+1} Hyperbolic Gradient Flow: Evolution of Graphs in R^{n+1}

2010-09-21

arxiv.org/abs/1009.3993v1

Mathematics

Differential Geometry

16 pages, 3 figures, This paper was accepted by the proceeding of the conference in honor of Professers Zhengguo Bai and Yibin

Scientific paper

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.

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