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Noncompact self-shrinkers for mean curvature flow with arbitrary genus

Reto Buzano, Huy The Nguyen, Mario B. Schulz

Abstract

In his lecture notes on mean curvature flow, Ilmanen conjectured the existence of noncompact self-shrinkers with arbitrary genus. Here, we employ min-max techniques to give a rigorous existence proof for these surfaces. Conjecturally, the self-shrinkers that we obtain have precisely one (asymptotically conical) end. We confirm this for large genus via a precise analysis of the limiting object of sequences of such self-shrinkers for which the genus tends to infinity. Finally, we provide numerical evidence for a further family of noncompact self-shrinkers with odd genus and two asymptotically conical ends.

Noncompact self-shrinkers for mean curvature flow with arbitrary genus

Abstract

In his lecture notes on mean curvature flow, Ilmanen conjectured the existence of noncompact self-shrinkers with arbitrary genus. Here, we employ min-max techniques to give a rigorous existence proof for these surfaces. Conjecturally, the self-shrinkers that we obtain have precisely one (asymptotically conical) end. We confirm this for large genus via a precise analysis of the limiting object of sequences of such self-shrinkers for which the genus tends to infinity. Finally, we provide numerical evidence for a further family of noncompact self-shrinkers with odd genus and two asymptotically conical ends.

Paper Structure

This paper contains 8 sections, 12 theorems, 4 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. Existence of noncompact self-shrinkers with given genus
  4. Construction of the sweepout
  5. Min-max procedure
  6. Determining the genus
  7. Behaviour for high genus
  8. Visualisations and numerical results

Key Result

Theorem 1.2

For all $g\in\mathbb{N}$ there exists a complete, embedded, noncompact self-shrinker $\Theta_g\subset\mathbb{R}^3$ for mean curvature flow which has genus $g$ and is invariant under the action of the dihedral group $\mathbb{D}_{g+1}$.

Figures (3)

  • Figure 3: Self-shrinkers of genus $g\in\{1,2,3\}$ with one end.
  • Figure 4: Self-shrinkers of genus $g\in\{4,5,11\}$ with one end and their vertical cuts containing the horizontal axes $\xi_1,\ldots,\xi_{g+1}$.
  • Figure 5: Top: Vertical cut through self-shrinkers of genus $13$ and $47$ with two ends. Bottom: Vertical cut through compact self-shrinkers of genus $14$ and $48$.

Theorems & Definitions (15)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Conjecture 1.4
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3: Stability of the Gaußian isoperimetric inequality
  • Lemma 2.4: cf. Carlotto2020
  • Lemma 2.5: Width estimate
  • Lemma 2.6
  • ...and 5 more