Meagerness of the realizable set for hypersurfaces of Euclidean four-space
Year: 2026
Type: preprint
Abstract: **Abstract.** Let $\mathcal U$ be the set of $C^{k,\alpha}$ Riemannian metrics on $S^3$ with $\det\mathbb M > 0$, where $\mathbb M = -G$ is the negative Einstein endomorphism, and let $F(g) = \mathrm{Cod}(\tilde K(g))$ with $\tilde K = \sqrt{\det\mathbb M}\,\mathbb M^{-1}$, so that $\mathcal R = F^{-1}(0)$ is the set of metrics realizable as isometrically immersed hypersurfaces of $E^4$. We prove that $\mathcal R$ is nowhere dense, hence meager, in a neighbourhood of the round metric. The proof is unconditional: it uses neither a slice theorem nor pseudodifferential machinery, only a uniform lower bound for $DF$ on a fixed transverse-traceless subspace. Separately, we show by an elementary argument that a closed hypersurface of $E^4$ satisfies $\det\mathbb M > 0$ somewhere, so that no compact three-manifold with $\det\mathbb M \equiv 0$ is realizable; this excludes six homogeneous geometries at once, among them the flat three-torus. We also compute the second variation of the total absolute curvature at the round sphere and determine its sign in each sector of the York decomposition. **MSC 2020.** 53C24 (primary); 53C42, 53A07, 58D17. **Keywords.** isometric embedding, hypersurfaces, Codazzi equation, Baire category, total absolute curvature.
Author: Thrussell, Jay
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204698706