Minimal immersions of closed surfaces in hyperbolic three-manifolds

TL;DR

This paper investigates minimal immersions of closed surfaces into hyperbolic 3-manifolds, revealing existence, non-existence, and curvature behavior depending on a parameter, with implications for geometric structures.

Contribution

It establishes existence and non-existence results for minimal immersions with prescribed data, and analyzes their curvature properties as a parameter varies.

Findings

Existence of at least two minimal immersions for small parameter values.

Non-existence of such immersions for sufficiently large parameter values.

Asymptotic curvature behavior as the parameter approaches zero.

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 $\alpha dz^2$ 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.