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

Primarily AI-generated textHuman understanding: no partsmath.DG — Differential Geometrymath.MG — Metric Geometry

Contributed by Jay Thrussell

Version 2 / Oct 31, 2026 / CC BY 4.0

Abstract

## Abstract Let (Σ,g)(\Sigma, g) be a closed Riemannian three-manifold, and let M=−G\mathbb M = -G be its negative Einstein endomorphism. Where det⁡M>0\det\mathbb M > 0, the companion paper [MA] reduces realizability as a hypersurface of E4E^4 to one equation, F(g)=0F(g) = 0, and studies it locally. Here we treat the global questions. - **Compactness.** A closed realization has det⁡M≥R−6\det\mathbb M \geq R^{-6} at a point farthest from any centre, where RR is that distance, and so max⁡det⁡M≥125/(8 diam6)\max\det\mathbb M \geq 125/(8\,\mathrm{diam}^6) by Jung's theorem. If det⁡M>0\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⁡M>0\det\mathbb M > 0 is realizable iff Σ\Sigma is simply connected and F(g)=0F(g) = 0. No closed three-manifold other than S3S^3 carries such a realizable metric. - **The homogeneous geometries.** A realizable closed metric with det⁡M>0\det\mathbb M > 0 and constant scalar curvature is round, and so is every compact locally homogeneous realization. Locally, constant M\mathbb M-eigenvalues with det⁡M>0\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)=∫det⁡M dVJ(g) = \int\sqrt{\det\mathbb M}\,dV equals 2π22\pi^2 on every closed realizable metric with det⁡M>0\det\mathbb M > 0, so any other value excludes realizability. At the round sphere JJ is stationary. Its second variation on transverse-traceless tensors of level nn is −(μn+2)(μn+3)/16-(\mu_n + 2)(\mu_n + 3)/16 times the squared norm, μn=n2−3\mu_n = n^2 - 3, and its radical is exactly the conformal and Lie directions. On conformal rays JJ is flat to third order. Along the conformal path of [MA], J=2π2−π2t4/12+O(t6)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+εcos⁡2χr = 1 + \varepsilon\cos2\chi, convexity is lost at ε=1/5\varepsilon = 1/5 in a Whitney cusp of the Gauss map. With δ=ε−1/5\delta = \varepsilon - 1/5, the relative excess J/2π2−1J/2\pi^2 - 1 then grows as 500 δ3/2/(92 π)500\,\delta^{3/2}/(9\sqrt2\,\pi). - **Connectivity, and the round sphere.** Modulo isometry, the closed realizable metrics with det⁡M>0\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.

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Version history

  1. v1Submitted by Jay ThrussellInitial depositSupersededOct 30, 2026
  2. v2Submitted by Jay ThrussellSubmitted as a new versionCurrentNov 02, 2026