Canonical foliation of bubblesheets

TL;DR

This paper introduces a new curvature condition called quasi-parallel mean curvature (QPMC) for high-codimension submanifolds, and demonstrates that certain geometric flow regions, called bubblesheets, can be canonically foliated by spheres with this property.

Contribution

The paper defines QPMC, broadens the class of submanifolds with known curvature properties, and establishes a canonical foliation for bubblesheets in nearly standard product manifolds.

Findings

High-curvature regions called bubblesheets can be normalized in a canonical form.

Manifolds close to $ ^k imes S^{n-k}$ admit a foliation by spheres with QPMC.

QPMC includes all CMC hypersurfaces and submanifolds with parallel mean curvature.

Abstract

We introduce a new curvature condition for high-codimension submanifolds of a Riemannian ambient space, called quasi-parallel mean curvature (QPMC). The class of submanifolds with QPMC includes all CMC hypersurfaces and submanifolds with parallel mean curvature. We use our notion of QPMC to prove that certain kinds of high-curvature regions which appear in geometric flows, called bubblesheets, can be placed in a suitable normal form. This follows from a more general result asserting that the manifold $\mathbb{R}^k \times \mathbb{S}^{n-k}$, equipped with any metric which is sufficiently close to the standard one, admits a canonical foliation by embedded $(n-k)$-spheres with QPMC.