PreprintCC BY 4.0arXiv
A Smooth Nonnegatively Curved Sphere Whose Zero-Curvature Point Locus Has Hausdorff Dimension 1/2
Abstract
For every integer m≥ 3, we construct a smooth Riemannian metric g on a manifold diffeomorphic to Sm such that secg≥ 0, every sectional curvature is strictly positive away from a closed nowhere-dense set, and the point locus on which some sectional curvature vanishes has Hausdorff dimension exactly 1/2. The metric is induced on the boundary of a smooth convex body in ℝm+1. The essential local model is a convex graph whose Hessian has a one-dimensional kernel precisely on a Cantor subset of one coordinate axis; a regularized maximum then inserts this graph into a round sphere without introducing additional degenerate points. The main content of this paper is generated by ChatGPT 5.6 and verified by the author.
Related
- Scalar curvature growth on nonnegatively curved three-manifolds
- A Model-threshold Dimension Bound and Sharp Critical Ends for Smooth Singular Sets of Constant Positive σk-curvature Metrics
- Valeur exacte de la mesure de Hausdorff d'un sous-ensemble fractal de dimension log 3 / log 4 du triangle de Sierpinski
- Effective Hausdorff dimension one is cofinal in the K-degrees
- On the Hausdorff dimension of oscillatory motions in three-body problems
BibTeX
@misc{liu2026smooth,
title = {{A Smooth Nonnegatively Curved Sphere Whose Zero-Curvature Point Locus Has Hausdorff Dimension 1/2}},
author = {Yuhang Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27898},
eprint = {2609.27898},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27898},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Smooth Nonnegatively Curved Sphere Whose Zero-Curvature Point Locus Has Hausdorff Dimension 1/2
AU - Yuhang Liu
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.27898
UR - https://arxiv.org/abs/2609.27898
AB - For every integer m≥ 3, we construct a smooth Riemannian metric g on a manifold diffeomorphic to S^m such that sec_g≥ 0, every sectional curvature is strictly positive away from a closed nowhere-dense set, and the point locus on which some sectional curvature vanishes has Hausdorff dimension exactly 1/2. The metric is induced on the boundary of a smooth convex body in ℝ^m+1. The essential local model is a convex graph whose Hessian has a one-dimensional kernel precisely on a Cantor subset of one coordinate axis; a regularized maximum then inserts this graph into a round sphere without introducing additional degenerate points. The main content of this paper is generated by ChatGPT 5.6 and verified by the author.
ER -