Closed three-manifolds as hypersurfaces of Euclidean four-space: 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 strictly convex embedded sphere, unique up to rigid motion. So a closed connected metric with $\det\mathbb M > 0$ is realizable iff $\Sigma$ is simply connected and $F(g) = 0$. No closed three-manifold other than $S^3$ carries such a realizable metric. - **The homogeneous geometries.** A realizable closed metric with $\det\mathbb M > 0$ and constant scalar curvature is round, and so is every compact locally homogeneous realization. Locally, constant $\mathbb M$-eigenvalues with $\det\mathbb M > 0$ force constant curvature, a case of Tsukada's classification. This recovers Agaoka and Hashinaga's list of the left-invariant metrics that embed locally. The eight Thurston geometries are worked as checks. - **Total absolute curvature.** $J(g) = \int\sqrt{\det\mathbb M}\,dV$ equals $2\pi^2$ on every closed realizable metric with $\det\mathbb M > 0$, so any other value excludes realizability. At the round sphere $J$ is stationary. Its second variation on transverse-traceless tensors of level $n$ is $-(\mu_n + 2)(\mu_n + 3)/16$ times the squared norm, $\mu_n = n^2 - 3$, and its radical is exactly the conformal and Lie directions. On conformal rays $J$ is flat to third order. Along the conformal path of [MA], $J = 2\pi^2 - \pi^2t^4/12 + O(t^6)$. - **Non-convex realizations.** Embedded hypersurfaces of revolution are non-convex exactly around their necks. For the radial graphs $r = 1 + \varepsilon\cos2\chi$, convexity is lost at $\varepsilon = 1/5$ in a Whitney cusp of the Gauss map. With $\delta = \varepsilon - 1/5$, the relative excess $J/2\pi^2 - 1$ then grows as $500\,\delta^{3/2}/(9\sqrt2\,\pi)$. - **Connectivity, and the round sphere.** Modulo isometry, the closed realizable metrics with $\det\mathbb M > 0$ form a contractible family, contracted by an explicit equivariant path. The embedded hypersurfaces of revolution join the round sphere through the degenerate locus. We collect the characterisations of the round sphere, with their hypotheses. Every displayed result is reproduced by a script; Appendix V indexes them.