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

TL;DR

This paper introduces the hyperbolic gradient flow for graphs in higher-dimensional space, analyzing its effects on convex hypersurfaces and plane curves, and demonstrating convergence to simpler geometric objects.

Contribution

It presents a novel geometric flow, the hyperbolic gradient flow, and studies its global behavior and convergence properties for manifolds and curves.

Findings

Convex hypersurfaces converge to hyperplanes.

Plane curves evolve to straight lines.

Hyperbolic conservation laws apply to manifold surgery.

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.