Slicing a 2-sphere

Abstract

We show that for every complete Riemannian surface $M$ diffeomorphic to a sphere with $k \geq 0$ holes there exists a Morse function $f:M \rightarrow \mathbb{R}$, which is constant on each connected component of the boundary of $M$ and has fibers of length no more than $52 \sqrt{Area(M)}+length(\partial M)$. We also show that on every 2-sphere there exists a simple closed curve of length $\leq 26 \sqrt{Area(S^2)}$ subdividing the sphere into two discs of area $\geq \frac{1}{3}Area(S^2)$