Rigidity of riemannian manifolds containing an equator

Laurent Mazet

doi:10.4310/CAG.240905212306

Rigidity of riemannian manifolds containing an equator

Differential Geometry 2024-12-24 v3

Authors: Laurent Mazet

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Abstract

In this paper, we prove that a Riemannian $n$ -manifold $M$ with sectional curvature bounded above by $1$ that contains a minimal $2$ -sphere of area $4\pi$ which has index at least $n-2$ has constant sectional curvature $1$ . The proof uses the construction of ancient mean curvature flows that flow out of a minimal submanifold. As a consequence we also prove a rigidity result for the Simon-Smith minimal spheres.

Keywords

riemannian manifold sectional curvature minimal sphere rigidity mean curvature flow index

Cite

@article{arxiv.2010.01994,
  title  = {Rigidity of riemannian manifolds containing an equator},
  author = {Laurent Mazet},
  journal= {arXiv preprint arXiv:2010.01994},
  year   = {2024}
}

Comments

33 pages, 1 figure. This new version contains a major improvement: the main novelty is a rigidity result for the min-max minimal spheres constructed by Simon and Smith

Rigidity of riemannian manifolds containing an equator

Abstract

Keywords

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