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Minimal immersions of closed surfaces in hyperbolic three-manifolds

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Abstract

We study minimal immersions of closed surfaces (of genus g ≥ 2) in hyperbolic three-manifolds, with prescribed data (σ, t α), where σ is a conformal structure on a topological surface S, and α dz 2 is a holomorphic quadratic differential on the surface (S, σ). We show that, for each \({t \in (0,\tau_0)}\) for some τ 0 > 0, depending only on (σ, α), there are at least two minimal immersions of closed surface of prescribed second fundamental form Re(t α) in the conformal structure σ. Moreover, for t sufficiently large, there exists no such minimal immersion. Asymptotically, as t → 0, the principal curvatures of one minimal immersion tend to zero, while the intrinsic curvatures of the other blow up in magnitude.

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  1. College of Staten Island, The City University of New York, Staten Island, NY, 10314, USA

    Zheng Huang & Marcello Lucia

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  1. Zheng Huang
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  2. Marcello Lucia
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Correspondence to Zheng Huang.

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Huang, Z., Lucia, M. Minimal immersions of closed surfaces in hyperbolic three-manifolds. Geom Dedicata 158, 397–411 (2012). https://doi.org/10.1007/s10711-011-9641-9

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