Minimal area surfaces and fibered hyperbolic $3$-manifolds

TL;DR

This paper demonstrates that in certain fibered hyperbolic 3-manifolds, the maximum principal curvature of least area minimal surfaces has a uniform lower bound exceeding one, extending previous curvature bounds.

Contribution

It provides a new, concise argument establishing a uniform lower bound for principal curvatures in specific fibered hyperbolic 3-manifolds, beyond Uhlenbeck's initial bounds.

Findings

Uniform lower bound for principal curvatures greater than one

Applicable to certain families of fibered hyperbolic 3-manifolds

Extends curvature bounds in minimal surface theory

Abstract

By work of Uhlenbeck, the largest principal curvature of any least area fiber of a hyperbolic $3$-manifold fibering over the circle is bounded below by one. We give a short argument to show that, along certain families of fibered hyperbolic $3$-manifolds, there is a uniform lower bound for the maximum principal curvatures of a least area minimal surface which is greater than one.