Minimal immersions of closed surfaces in hyperbolic three-manifolds

Abstract

We study minimal immersions of closed surfaces (of genus $g\ge 2$) in hyperbolic 3-manifolds, with prescribed data $(\sigma ,t\alpha )$, where $\sigma$ is a conformal structure on a topological surface $S$, and $\alpha d{z}^2$ is a holomorphic quadratic differential on the surface $(S,\sigma )$. We show that, for each $t\in (0,{\tau }_0)$ for some ${\tau }_0>0$, depending only on $(\sigma ,\alpha )$, there are at least two minimal immersions of closed surface of prescribed second fundamental form $Re(t\alpha )$ in the conformal structure $\sigma$. Moreover, for $t$ sufficiently large, there exists no such minimal immersion. Asymptotically, as $t\to 0$, the principal curvatures of one minimal immersion tend to zero, while the intrinsic curvatures of the other blow up in magnitude.

16 pages

References in corpus (2)

• On the renormalized volume of hyperbolic 3-manifolds
• Minimal surfaces and particles in 3-manifolds

Cited by in corpus (2)

• Bifurcation for Minimal Surface Equation in Hyperbolic $3$-Manifolds
• Complex length of short curves and Minimal Fibration in hyperbolic $3$-Manifolds fibering over the circle