Principal curvatures of fibers and Heegaard surfaces

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.