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Classification of quadratically pinched self-shrinkers in higher codimension

Debora Impera, Michele Rimoldi, Francesco Ruatta

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.

Classification of quadratically pinched self-shrinkers in higher codimension

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 and hence to sharpen, in the self-shrinker setting, the pinching constants appearing in recent classification results for ancient solutions.
Paper Structure (13 sections, 10 theorems, 84 equations)

This paper contains 13 sections, 10 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Geometric setup and notation
  3. The second fundamental form
  4. Fundamental equations
  5. Weighted manifolds, self-shrinkers, and weighted parabolicity
  6. Simons' equations
  7. Simons' equation for |H|2
  8. Simons' equation for |H| with parallel principal normal
  9. Simons' equation for |B|2
  10. Simons' equation for |BH/|H|| with parallel principal normal
  11. Two lower bounds for Delta-f |B|2
  12. Proof of Theorem \ref{['thm:extensionsmoczyck']}
  13. Proof of the main result

Key Result

Theorem 1.1

Let $x:M^n\to\mathbb{R}^{n+1}$ be a complete immersed self-shrinker without boundary. If $M$ has polynomial volume growth and nonnegative mean curvature ($H\geq 0$), then it is isometric either to the product $\Gamma\times \mathbb R^{n-1}$, where $\Gamma$ is an Abresch-Langer curve, or to $\mathbb S

Theorems & Definitions (15)

  • Theorem 1.1: colding_minicozzi_2012_generic_singularities, Theorem 0.17
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Lemma 3.1: anmin_li_jimin_li_1992, Theorem 1
  • Lemma 3.2: cheng_2002, Lemma 3.2
  • Proposition 3.3
  • proof
  • Theorem 4.1: smoczyk_2005, Theorem 1.2
  • Lemma 4.2
  • ...and 5 more