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

TL;DR

This paper investigates the Calabi-Yau conjectures for minimal hypersurfaces in higher dimensions, proving a chord-arc estimate for bounded curvature cases and constructing counterexamples to properness in dimensions three and above.

Contribution

It provides a new chord-arc estimate for minimal disks with bounded curvature and constructs explicit counterexamples to the properness conjecture in higher dimensions.

Findings

Chord-arc estimate for minimal disks with bounded curvature

Existence of improperly embedded minimal hypersurfaces in higher dimensions

Counterexamples to the properness conjecture in dimensions n ≥ 3

Abstract

In this paper, we study the Calabi-Yau conjectures for complete minimal hypersurfaces $\Sigma^{n}\subset \mathbb{R}^{n+1}$ 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.