Principal curvatures of fibers and Heegaard surfaces

William Breslin

Vol. 250 (2011), No. 1, 61–66

DOI: 10.2140/pjm.2011.250.61

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 δ > 0 such that the fiber or Heegaard surface must contain a point at which one of the principal curvatures exceeds 1 + δ in absolute value.

Keywords

hyperbolic manifold, Heegaard surface, fiber, principal curvatures

Mathematical Subject Classification 2000

Primary: 57M50

Milestones

Received: 3 May 2010

Accepted: 17 August 2010

Published: 1 March 2011