A Smooth Nonnegatively Curved Sphere Whose Zero-Curvature Point Locus Has Hausdorff Dimension 1/2
Year: 2026
Type: preprint
Abstract: For every integer $m\geq 3$, we construct a smooth Riemannian metric $g$ on a manifold diffeomorphic to $S^m$ such that $\text{sec}_g\geq 0$, 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 $\frac{1}{2}$. The metric is induced on the boundary of a smooth convex body in $\mathbb{R}^{m+1}$. 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.
Source: arXiv (Cornell University)
Author: Liu, Yuhang
Institution: Xi’an Jiaotong-Liverpool University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214205154