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Nonproduct Minimal Submanifolds with Cylindrical Tangent Cones
Abstract
This article constructs, via perturbation methods, infinite-dimensional families of non-product minimal submanifolds which have singular sets locally modeled on cylindrical tangent cones. This answers a question raised by Leon Simon in the 1980s, and generalizes work Caffarelli-Hardt-Simon, who treated the case of isolated singularities. This setting requires substantially more work, however, since the natural obstruction space for the problem is infinite dimensional and the map which is meant to compensate for this obstruction using deformations of the singular set exhibits a loss of regularity. This necessitates working in the category of tame Frechet spaces and using the Nash-Moser Implicit Function Theorem.
Related
- Uniqueness of certain cylindrical tangent cones
- Uniqueness of tangent cones with immersed special Legendrian links
- Scalar curvature bounds and tangent cones for Kähler--Ricci shrinkers
- Rigidity of Minimal Submanifolds in Spheres of Higher Codimension
- Orientable minimal submanifolds of low index in the sphere
BibTeX
@misc{mazzeo2026nonproduct,
title = {{Nonproduct Minimal Submanifolds with Cylindrical Tangent Cones}},
author = {Rafe Mazzeo and Gregory J. Parker},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20862},
eprint = {2610.20862},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20862},
}
RIS
TY - UNPB
TI - Nonproduct Minimal Submanifolds with Cylindrical Tangent Cones
AU - Rafe Mazzeo
AU - Gregory J. Parker
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20862
UR - https://arxiv.org/abs/2610.20862
AB - This article constructs, via perturbation methods, infinite-dimensional families of non-product minimal submanifolds which have singular sets locally modeled on cylindrical tangent cones. This answers a question raised by Leon Simon in the 1980s, and generalizes work Caffarelli-Hardt-Simon, who treated the case of isolated singularities. This setting requires substantially more work, however, since the natural obstruction space for the problem is infinite dimensional and the map which is meant to compensate for this obstruction using deformations of the singular set exhibits a loss of regularity. This necessitates working in the category of tame Frechet spaces and using the Nash-Moser Implicit Function Theorem.
ER -