We consider a problem of subdividing and sweeping out Riemannian 2-surfaces by short curves and cycles.
We prove that for every Riemannian 2-disc or 2-sphere $M$ there exists a sweepout of $M$ by based loops whose length is bounded in terms of area, diameter and boundary lenght of $M$. We provide nearly optimal bounds for the lengths of curves in a sweepout and use it to obtain curvature-free upper bounds for lengths of simple closed geodesics on a 2-sphere. This is a joint work with Alexander Nabutovsky and Regina Rotman.
In the third chapter we construct a sequence of metrics on a closed surface $M$, such that $M$ has small diameter, but the maximal length of a curve or 1-cycle in an optimal sweepout of $M$ is arbitrarily large.
In the fourth chapter we show that every Riemannian surface $M$ diffeomorphic to a sphere with $k \geq 0$ holes admits a Morse function whose fibers are bounded by
$52 \sqrt{Area(M)}+length(\partial M)$. 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 $\geq \frac{1}{3}Area(S^2)$.
In the fifth chapter we consider a problem of sweeping out a closed surface $M$ by high parametric families of short 1-cycles. We construct a family of 1-cycles with $\mathbb{Z}_2$ coefficients on $M$ that represents a non-trivial element of the k'th homology group of the space of cycles and such that the mass of each cycle is bounded above by $C \max\{\sqrt{k}, \sqrt{g}\} \sqrt{Area(M)}$. This result is optimal up to a multiplicative constant.
In the last chapter we show that if there exists a sweepout of a Riemannian 2-sphere $M$ by curves of length less than $L$ then it is possible to construct a slicing of $M$ by simple curves which do not intersect and such that their length is also bounded by $L$. This is a joint work with Gregory R. Chambers.