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Existence of Minimal Surfaces in Infinite Volume Hyperbolic 3-manifolds
[Submitted on 19 Feb 2025 (v1), last revised 12 Jun 2026 (this v · 2026-06-15 · via math updates on arXiv.org

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Abstract:The existence of embedded minimal surfaces in non-compact 3-manifolds remains a largely unresolved and challenging problem in geometry. In this paper, we address several open cases regarding the existence of finite-area, embedded, complete, minimal surfaces in infinite-volume hyperbolic 3-manifolds.
Among other results, for doubly degenerate manifolds with bounded geometry, we prove a dichotomy: either every such manifold contains a closed minimal surface or there exists such a manifold admitting a foliation by closed minimal surfaces. We also construct the first examples of Schottky manifolds with closed minimal surfaces and demonstrate the existence of Schottky manifolds containing infinitely many closed minimal surfaces. Lastly, for hyperbolic 3-manifolds with rank-1 cusps, we show that a broad class of these manifolds must contain a finite-area, embedded, complete minimal surface.

Submission history

From: Baris Coskunuzer [view email]
[v1] Wed, 19 Feb 2025 15:00:30 UTC (153 KB)
[v2] Sat, 14 Jun 2025 09:51:18 UTC (124 KB)
[v3] Fri, 12 Jun 2026 12:02:51 UTC (115 KB)

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