Classification of quadratically pinched self-shrinkers in higher codimension

TL;DR

This paper classifies certain self-shrinkers in mean curvature flow with quadratic pinching conditions, showing they are essentially generalized cylinders, using a novel elliptic approach that sharpens existing classification results.

Contribution

It introduces a purely elliptic method to classify self-shrinkers under quadratic pinching, extending results to higher codimension without uniform pinching assumptions.

Findings

Self-shrinkers reduce to codimension one under the pinching condition

The classification applies in any dimension with the sharp Andrews-Baker constant

The approach sharpens classification results for ancient solutions

Abstract

We classify properly immersed self-shrinkers of the mean curvature flow in arbitrary codimension under a quadratic pinching condition of Andrews-Baker type on the second fundamental form that is preserved along the flow. Under this assumption, such self-shrinkers reduce effectively to codimension one and are therefore generalized self-shrinking cylinders. In contrast to previous works, our approach is purely elliptic: it relies on parabolicity in a weighted setting and is tailored specifically to self-shrinkers, rather than to general ancient solutions of the flow. This allows us to avoid assuming any uniform pinching condition, to treat in any dimension the sharp Andrews-Baker pinching constant $\frac{4}{3n}$ and hence to sharpen, in the self-shrinker setting, the pinching constants appearing in recent classification results for ancient solutions.