Observing minimal submanifolds with isolated singularities on open sets of metrics
Abstract
We develop a perturbative theory for minimal submanifolds with only (strongly) isolated singularities, with respect to varying background Riemannian metrics. As an application, we obtain a general "sporadic persistence" criterion, thus naturally complementing the non-persistence theorem by the authors. In particular, such a criterion is satisfied by the spherical suspension in π3m+2 of the Veronese embedding of πβ2 for π=β,β,β,π (the normed division algebras) in the round spheres of dimension 3m+1=4,7,13,25 respectively, thereby providing families of continuously varying (2m+1)-dimensional stationary varifolds with isolated singularities that are observable on an open set of metrics. By our previous results, these phenomena cannot possibly occur in the codimension one scenario.
Related
- Complete conformal metrics of constant positive scalar curvature with prescribed isolated singularities
- Removability of non-isolated singularities for Einstein metrics and RCD spaces
- Rigidity of Minimal Submanifolds in Spheres of Higher Codimension
- Orientable minimal submanifolds of low index in the sphere
- The pinching constant for closed minimal submanifolds of high codimension in the sphere
BibTeX
@misc{carlotto2026observing,
title = {{Observing minimal submanifolds with isolated singularities on open sets of metrics}},
author = {Alessandro Carlotto and Yangyang Li and Zhihan Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22229},
eprint = {2610.22229},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22229},
}
RIS
TY - UNPB
TI - Observing minimal submanifolds with isolated singularities on open sets of metrics
AU - Alessandro Carlotto
AU - Yangyang Li
AU - Zhihan Wang
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.22229
UR - https://arxiv.org/abs/2610.22229
AB - We develop a perturbative theory for minimal submanifolds with only (strongly) isolated singularities, with respect to varying background Riemannian metrics. As an application, we obtain a general "sporadic persistence" criterion, thus naturally complementing the non-persistence theorem by the authors. In particular, such a criterion is satisfied by the spherical suspension in π^3m+2 of the Veronese embedding of πβ^2 for π=β,β,β,π (the normed division algebras) in the round spheres of dimension 3m+1=4,7,13,25 respectively, thereby providing families of continuously varying (2m+1)-dimensional stationary varifolds with isolated singularities that are observable on an open set of metrics. By our previous results, these phenomena cannot possibly occur in the codimension one scenario.
ER -