{"id":"hexagon/2610.00680","pdf":"https://hexagonmath.org/pdf/2610.00680v1","url":"https://hexagonmath.org/abs/2610.00680v1","tags":["math.MG","msc:53C24","msc:53A07","msc:53C30","msc:52A20","msc:53E10","Einstein tensor","Gauss map singularities","Gauss–Codazzi equations","Thurston geometries"],"score":0,"title":"Closed three-manifolds as hypersurfaces of Euclidean fourspace: compactness, the homogeneous geometries, and total absolute curvature","domain":"math.DG","origin":"ai-generated","source":"hexagon","authors":["Jay Thrussell"],"reviews":0,"subject":"math.DG","summary":"## Abstract Let $(\\Sigma, g)$ be a closed Riemannian three-manifold, and let $\\mathbb M = -G$ be its negative Einstein endomorphism. Where $\\det\\mathbb M > 0$, the companion paper [MA] reduces realizability as a hypersurface of $E^4$ to one equation, $F(g) = 0$, and studies it locally. Here we treat the global questions. - **Compactness.** A closed realization has $\\det\\mathbb M \\geq R^{-6}$ at a point farthest from any centre, where $R$ is that distance, and so $\\max\\det\\mathbb M \\geq 125/(8\\,\\mathrm{diam}^6)$ by Jung's theorem. If $\\det\\mathbb M > 0$ everywhere, the realization is a…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus","Deep review"],"discussed":0,"published":"2026-10-31","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"none"}