On Cylindrical singularities of minimal hypersurfaces
Abstract
In a neighborhood of the origin in ℝ9, we construct a minimal hypersurface with respect to a smooth conformally Euclidean metric, whose singular set consists of the origin and a sequence of isolated singular points converging to it. At each of the isolated singular points the hypersurface has the unique tangent cone C4,3, the cone over S4(√(4/7))× S3(√(3/7)), whereas at the origin its unique tangent cone is the cone C3,3×ℝ. The metric is Euclidean near each isolated singular point and agrees with the Euclidean metric to infinite order at the origin. A key step, of independent interest, is the construction of an entire stationary hypersurface in Euclidean ℝ9 with exactly one singular point at the origin, at which its unique tangent cone is C4,3, and whose unique tangent cone at infinity is C3,3×ℝ.
Related
- Mean curvature flows with cylindrical singularities II: stability and genericity
- Morse resolution of mean curvature flows with cylindrical singularities
- Singularities of totally invariant hypersurfaces of endomorphisms
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- Nonproduct Minimal Submanifolds with Cylindrical Tangent Cones
BibTeX
@misc{halavati2026cylindrical,
title = {{On Cylindrical singularities of minimal hypersurfaces}},
author = {Aria Halavati and Luca Spolaor},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10754},
eprint = {2610.10754},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10754},
}
RIS
TY - UNPB
TI - On Cylindrical singularities of minimal hypersurfaces
AU - Aria Halavati
AU - Luca Spolaor
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10754
UR - https://arxiv.org/abs/2610.10754
AB - In a neighborhood of the origin in ℝ^9, we construct a minimal hypersurface with respect to a smooth conformally Euclidean metric, whose singular set consists of the origin and a sequence of isolated singular points converging to it. At each of the isolated singular points the hypersurface has the unique tangent cone C_4,3, the cone over S^4(√(4/7))× S^3(√(3/7)), whereas at the origin its unique tangent cone is the cone C_3,3×ℝ. The metric is Euclidean near each isolated singular point and agrees with the Euclidean metric to infinite order at the origin. A key step, of independent interest, is the construction of an entire stationary hypersurface in Euclidean ℝ^9 with exactly one singular point at the origin, at which its unique tangent cone is C_4,3, and whose unique tangent cone at infinity is C_3,3×ℝ.
ER -