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Minimal 2-Spheres and Optimal Foliations in 3-Spheres with Arbitrary Metric
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Minimal $2$-Spheres and Optimal Foliations in $3$-Spheres with Arbitrary Metric
abstract
In this paper, we prove that the $3$-sphere endowed with an arbitrary Riemannian metric either contains at least two embedded minimal $2$-spheres or admits an optimal foliation by $2$-spheres. This generalizes recent results by Haslhofer-Ketover (Duke Math. J. 2019), where the existence of optimal foliations and minimal $2$-spheres has been established under the additional assumption that the metric is generic. In light of recent examples by Wang-Zhou, where min-max for some non-bumpy metrics on the 3-sphere produces higher multiplicities, our results are in a certain sense sharp.
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Cited by 1 Pith paper
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Existence of two embedded minimal spheres in $S^3$ with an arbitrary metric
Every Riemannian manifold diffeomorphic to S^{3} contains at least two distinct embedded minimal 2-spheres.
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