Wandering Singularities

TL;DR

This paper investigates the long-term behavior of parabolic geometric flows, showing that most singularities are avoided over time and that hypersurfaces tend to wander away from complex singularities, with special focus on the dynamics near spherical singularities.

Contribution

It demonstrates that, under certain conditions, generic initial hypersurfaces tend to leave neighborhoods of singularities and do not return, extending previous work on the dynamics of geometric flows.

Findings

Most singularities are avoided over long time.

Hypersurfaces tend to wander away from non-spherical singularities.

Near spherical singularities, hypersurfaces remain close and do not wander.

Abstract

Parabolic geometric flows are smoothing for short time however, over long time, singularities are typically unavoidable, can be very nasty and may be impossible to classify. The idea of [CM6] and here is that, by bringing in the dynamical properties of the flow, we obtain also smoothing for large time for generic initial conditions. When combined with [CM1], this shows, in an important special case, the singularities are the simplest possible. The question of the dynamics of a singularity has two parts. One is: What are the dynamics near a singularity? The second is: What is the long time behavior? That is, if the flow leaves a neighborhood of a singularity, can it return at a much later time? The first question was addressed in [CM6] and the second here. Combined with [CM1], [CM6], we show that all other closed singularities than the (round) sphere have a neighborhood where `nearly every' closed hypersurface leaves under the flow and never returns, even to a dilated, rotated or translated copy of the singularity. In other words, it wanders off. In contrast, by Huisken, any closed hypersurface near a sphere remains close to a dilated or translated copy of the sphere at each time.