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 $\geq \frac{ 1}{ 3}Area(S^2)$
Categories: math.DG