Existence of four minimal spheres in $S^3$ with a bumpy metric

Zhichao Wang; Xin Zhou

Existence of four minimal spheres in $S^3$ with a bumpy metric

Differential Geometry 2024-06-18 v2 Analysis of PDEs Geometric Topology

Authors: Zhichao Wang , Xin Zhou

Abstract

We prove that in the three dimensional sphere with a bumpy metric or a metric with positive Ricci curvature, there exist at least four distinct embedded minimal two-spheres. This confirms a conjecture of S. T. Yau in 1982 for bumpy metrics and metrics with positive Ricci curvature. The proof relies on a multiplicity one theorem for the Simon-Smith min-max theory.

Keywords

minimal spheres bumpy metric positive ricci curvature three dimensional sphere embedded minimal two-spheres yau conjecture simon-smith min-max theory multiplicity one theorem

Cite

@article{arxiv.2305.08755,
  title  = {Existence of four minimal spheres in $S^3$ with a bumpy metric},
  author = {Zhichao Wang and Xin Zhou},
  journal= {arXiv preprint arXiv:2305.08755},
  year   = {2024}
}

Comments

60 pages; minor revision: typo corrected, an appendix on the Lusternik-Schnirelmann inequality added, references updated; comments welcome

Existence of four minimal spheres in $S^3$ with a bumpy metric

Abstract

Keywords

Cite

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