English

Slicing a 2-sphere

Differential Geometry 2014-07-01 v2

Abstract

We show that for every complete Riemannian surface MM diffeomorphic to a sphere with k0k \geq 0 holes there exists a Morse function f:MRf:M \rightarrow \mathbb{R}, which is constant on each connected component of the boundary of MM and has fibers of length no more than 52Area(M)+length(M)52 \sqrt{Area(M)}+length(\partial M). We also show that on every 2-sphere there exists a simple closed curve of length 26Area(S2)\leq 26 \sqrt{Area(S^2)} subdividing the sphere into two discs of area 13Area(S2)\geq \frac{1}{3}Area(S^2)

Keywords

Cite

@article{arxiv.1401.0060,
  title  = {Slicing a 2-sphere},
  author = {Yevgeny Liokumovich},
  journal= {arXiv preprint arXiv:1401.0060},
  year   = {2014}
}

Comments

19 pages. Exposition improved. To be published in Journal of Topology and Analysis

R2 v1 2026-06-22T02:37:23.498Z