Closed three-manifolds as hypersurfaces of Euclidean fourspace: compactness, the homogeneous geometries, and total absolute curvature
## 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…
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