Original Paper

Geometriae Dedicata

, Volume 158, Issue 1, pp 397-411

Minimal immersions of closed surfaces in hyperbolic three-manifolds

  • Zheng HuangAffiliated withCollege of Staten Island, The City University of New York Email author 
  • , Marcello LuciaAffiliated withCollege of Staten Island, The City University of New York

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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.

Keywords

Minimal immersion Second fundamental form Mountain pass solution Hyperbolic three-manifolds

Mathematics Subject Classification (2000)

53C21 53A10 35J62