Principal curvatures of fibers and Heegaard surfaces

TL;DR

This paper proves that in hyperbolic 3-manifolds, fibers and Heegaard surfaces with bounded genus and injectivity radius must have points where principal curvatures exceed 1 by a positive margin, revealing geometric constraints.

Contribution

It establishes a quantitative lower bound on principal curvatures for fibers and Heegaard surfaces under specific geometric conditions, extending understanding of their curvature properties.

Findings

Existence of a positive delta such that principal curvatures exceed 1 + delta

Bounded genus and injectivity radius impose curvature constraints

Principal curvatures cannot be uniformly less than or equal to 1 in absolute value

Abstract

We study principal curvatures of fibers and Heegaard surfaces smoothly embedded in hyperbolic 3-manifolds. It is well known that a fiber or a Heegaard surface in a hyperbolic 3-manifold cannot have principal curvatures everywhere less than one in absolute value. We show that given an upper bound on the genus of a minimally embedded fiber or Heegaard surface and a lower bound on the injectivity radius of the hyperbolic 3-manifold, there exists a $\delta > 0$ such that the fiber or Heegaard surface must contain a point at which one of the principal curvatures is greater than $1 + \delta$ in absolute value.