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On the Calabi-Yau Conjectures for Minimal Hypersurfaces in Higher Dimensions

Shrey Aryan, Alexander McWeeney

Abstract

In this paper, we study the Calabi-Yau conjectures for complete minimal hypersurfaces $Σ^{n}\subset \mathbb{R}^{n+1}$ in dimensions $n\ge 3$. These conjectures ask whether a complete minimal hypersurface must be unbounded, and more strongly, whether it must be proper. For the unboundedness question, we prove a chord-arc estimate for an embedded minimal disk with bounded curvature, showing that intrinsic distance is controlled by a polynomial of the extrinsic distance. On the other hand, using gluing techniques, we construct a complete, improperly embedded minimal hypersurface in $\mathbb{R}^{n+1}$ for every $n\ge 3$. This example shows that the properness conjecture suggested by the deep work of Colding Minicozzi [CM08] in the case $n=2$ fails in higher dimensions.

On the Calabi-Yau Conjectures for Minimal Hypersurfaces in Higher Dimensions

Abstract

In this paper, we study the Calabi-Yau conjectures for complete minimal hypersurfaces in dimensions . These conjectures ask whether a complete minimal hypersurface must be unbounded, and more strongly, whether it must be proper. For the unboundedness question, we prove a chord-arc estimate for an embedded minimal disk with bounded curvature, showing that intrinsic distance is controlled by a polynomial of the extrinsic distance. On the other hand, using gluing techniques, we construct a complete, improperly embedded minimal hypersurface in for every . This example shows that the properness conjecture suggested by the deep work of Colding Minicozzi [CM08] in the case fails in higher dimensions.
Paper Structure (45 sections, 38 theorems, 217 equations)

This paper contains 45 sections, 38 theorems, 217 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Chord--Arc Estimates for Bounded Curvature
  4. Step 1
  5. Step 2
  6. Step 3
  7. Step 4
  8. Improperly Embedded Complete Minimal Hypersurface in $\mathbb{R}^{n +1}$
  9. Overview of the Construction
  10. Step 1
  11. Step 1(a)
  12. Step 1(b)
  13. Step 1(c)
  14. Step 2
  15. Step 2(a)
  16. ...and 30 more sections

Key Result

Theorem 1

Let $3\leq n\leq 5$, $\Sigma^n \subset \mathbb{R}^{n+1}$ be a two-sided, embedded minimal disk with $\sup_{\Sigma}|A| < \infty$. Then there exists a constant $c = c\bigl(n,\sup_{\Sigma}|A|\bigr) > 0$ with the following property: for every $x\in \Sigma$ and every $R>1$ such that the intrinsic ball $\ In particular, intrinsic distances are quantitatively controlled by extrinsic distances on $\Sigma_

Theorems & Definitions (89)

  • Conjecture 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 79 more