Bifurcation for minimal surface equation in hyperbolic 3-manifolds

Minimal surfaces in germs of hyperbolic 3–manifolds

This article introduces a universal moduli space for the set whose archetypal element is a pair that consists of a metric and second fundamental form from a compact, oriented, positive genus minimal surface in some hyperbolic 3–manifold. This moduli space is a smooth, finite dimensional manifold with canonical maps to both the cotangent bundle of the Teichmüller space and the space of SO3(C) re...

Minimal Surfaces in Finite Volume Hyperbolic 3-manifolds N and in M × S, M a Finite Area Hyperbolic Surface

We consider properly immersed finite topology minimal surfaces Σ in complete finite volume hyperbolic 3-manifolds N , and in M × S, where M is a complete hyperbolic surface of finite area. We prove Σ has finite total curvature equal to 2π times the Euler characteristic χ(Σ) of Σ, and we describe the geometry of the ends of Σ. .

The Minimal Volume Orientable Hyperbolic 2-cusped 3-manifolds

We prove that the Whitehead link complement and the (−2, 3, 8) pretzel link complement are the minimal volume orientable hyperbolic 3-manifolds with two cusps, with volume 3.66... = 4 × Catalan’s constant. We use topological arguments to establish the existence of an essential surface which provides a lower bound on volume and strong constraints on the manifolds that realize that lower bound.

Bifurcation of Minimal Surfaces in Riemannian Manifolds

We study the bifurcation of closed minimal surfaces in Riemannian manifolds through higher order variations of the area functional and relate it to elementary catastrophes.

Horocycles in hyperbolic 3-manifolds