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Sweeping out 3-manifold of positive Ricci curvature by short 1-cycles via estimates of min-max surfaces

Yevgeny Liokumovich South Kensington Campus Xin Zhou Massachusetts Institute of Technology

Differential Geometry mathscidoc:1610.10028

Mathematics, 2016.5
We prove that given a three manifold with an arbitrary metric $(M^3, g)$ of positive Ricci curvature, there exists a sweepout of $M$ by surfaces of genus $\leq 3$ and areas bounded by $C vol(M^3, g)^{2/3}$. We use this result to construct a sweepout of $M$ by 1-cycles of length at most $C vol(M^3, g)^{1/3}$ and prove a systolic inequality for all $M \neq S^3$. The sweepout of surfaces is generated from a min-max minimal surface. If further assuming a positive scalar curvature lower bound, we can get a diameter upper bound for the min-max surface.
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@inproceedings{yevgeny2016sweeping,
  title={Sweeping out 3-manifold of positive Ricci curvature by short 1-cycles via estimates of min-max surfaces},
  author={Yevgeny Liokumovich, and Xin Zhou},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161010104750649555107},
  booktitle={Mathematics},
  year={2016},
}
Yevgeny Liokumovich, and Xin Zhou. Sweeping out 3-manifold of positive Ricci curvature by short 1-cycles via estimates of min-max surfaces. 2016. In Mathematics. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161010104750649555107.
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