On the Calabi-Yau Conjectures for Minimal Hypersurfaces in Higher Dimensions
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2026
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| _version_ | 1866910034615599104 |
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| author | Aryan, Shrey McWeeney, Alexander D. |
| author_facet | Aryan, Shrey McWeeney, Alexander D. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_16048 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Calabi-Yau Conjectures for Minimal Hypersurfaces in Higher Dimensions Aryan, Shrey McWeeney, Alexander D. Differential Geometry Analysis of PDEs 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. |
| title | On the Calabi-Yau Conjectures for Minimal Hypersurfaces in Higher Dimensions |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2602.16048 |