Geometriae Dedicata

, Volume 158, Issue 1, pp 397–411

Minimal immersions of closed surfaces in hyperbolic three-manifolds

Original Paper

DOI: 10.1007/s10711-011-9641-9

Cite this article as:
Huang, Z. & Lucia, M. Geom Dedicata (2012) 158: 397. doi:10.1007/s10711-011-9641-9
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Abstract

We study minimal immersions of closed surfaces (of genus g ≥ 2) in hyperbolic three-manifolds, with prescribed data (σ, tα), where σ is a conformal structure on a topological surface S, and αdz2 is a holomorphic quadratic differential on the surface (S, σ). We show that, for each \({t \in (0,\tau_0)}\) for some τ0 > 0, depending only on (σ, α), there are at least two minimal immersions of closed surface of prescribed second fundamental form Re(tα) 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.

Keywords

Minimal immersionSecond fundamental formMountain pass solutionHyperbolic three-manifolds

Mathematics Subject Classification (2000)

53C2153A1035J62

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.College of Staten Island, The City University of New YorkStaten IslandUSA