Almost-fuchsian structures on disc bundles over a surface

Bronstein, Samuel

doi:10.2140/apde.2026.19.1361

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Almost-fuchsian structures on disc bundles over a surface

Samuel Bronstein

Vol. 19 (2026), No. 7, 1361–1394

DOI: 10.2140/apde.2026.19.1361

Abstract

Considering an integer $d > 0$ , we show the existence of convex-cocompact representations of surface groups into $SO (4, 1)$ admitting an embedded minimal map with small second fundamental form, and whose associated hyperbolic $4$ -manifolds are disc bundles of degree $d$ over the surface, provided its genus is large enough. We also show that we can realize these representations as complex variation of Hodge structures. This gives examples of quasicircles in $𝕊^{3}$ bounding superminimal discs in $ℍ^{4}$ of arbitrarily small second fundamental form. Those are examples of generalized almost-fuchsian representations which are not deformations of fuchsian representations by discrete and faithful representations.

Keywords

minimal surface, almost-fuchsian, hyperbolic, disc bundles

Mathematical Subject Classification

Primary: 51M10, 53A10, 53A35, 58J90

Milestones

Received: 10 August 2024

Revised: 14 July 2025

Accepted: 13 October 2025

Published: 18 September 2026

Authors

Samuel Bronstein
Max Planck Institut für Mathematik in der Wissenschaften Leipzig Germany

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Vol. 19, No. 7, 2026