Examples of hypersurfaces flowing by curvature in a Riemannian manifold

TL;DR

This paper explores specific examples of hypersurfaces evolving under curvature-driven flows in hyperbolic manifolds, highlighting convergence behaviors and the influence of topology on flow outcomes.

Contribution

It provides explicit examples of curvature flows, especially harmonic mean curvature flow, demonstrating convergence to geodesic submanifolds and analyzing effects of topology.

Findings

Hypersurfaces converge to totally geodesic submanifolds.

Flow by harmonic mean curvature can exist for infinite time.

Genus influences the evolution of harmonic mean curvature flow in surfaces.

Abstract

This paper gives some examples of hypersurfaces $\phi_t(M^n)$ evolving in time with speed determined by functions of the normal curvatures in an $(n+1)$-dimensional hyperbolic manifold; we emphasize the case of flow by harmonic mean curvature. The examples converge to a totally geodesic submanifold of any dimension from 1 to $n$, and include cases which exist for infinite time. Convergence to a point was studied by Andrews, and only occurs in finite time. For dimension $n=2,$ the destiny of any harmonic mean curvature flow is strongly influenced by the genus of the surface $M^2$.