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This is a Quantitative Study study.

September 30, 2025Open Access

Area bounds for constant mean curvature surfaces in hyperbolic 3-manifolds

Key Points

An upper bound on area exists for closed embedded surfaces with constant mean curvature at least one.
The area bound relates to the mean curvature and genus bounds, ensuring compactness for such surfaces.
Bryant surfaces with constant mean curvature equal to one have their area bounded proportionally to the genus.
These considerations are significant for understanding the geometry of hyperbolic 3-manifolds.

Abstract

On a finite-volume hyperbolic 3-manifold, we establish an upper bound on the area of closed embedded surfaces with constant mean curvature at least one, depending on the mean curvature and the genus bounds. This area bound implies compactness for such surfaces. In particular, for Bryant surfaces with constant mean curvature equal to one, the area is bounded proportionally to the genus.

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Cite This Study

Ruojing Jiang (Thu,) studied this question.

synapsesocial.com/papers/68dc1e308a7d58c25ebb1414 https://doi.org/https://doi.org/10.48550/arxiv.2509.09881

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