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

zbMATH Open1320.53080arXiv1009.3993MaRDI QIDQ5500027

De-Xing Kong, Kefeng Liu

Publication date: 5 August 2015

Abstract: 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.

Full work available at URL: https://arxiv.org/abs/1009.3993

zbMATH Keywords

global existenceshock wavesmooth solutiongeometry of manifoldhyperbolic gradient flow

Mathematics Subject Classification ID

First-order nonlinear hyperbolic equations (35L60) Methods of global Riemannian geometry, including PDE methods; curvature restrictions (53C21) Initial value problems for first-order hyperbolic systems (35L45) Hyperbolic equations on manifolds (58J45)

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