Density of minimal hypersurfaces for generic metrics.

By Kei Irie, F. C. Marques, A. Neves. 2017

107 citations recorded by the source corpus.

For almost all Riemannian metrics (in the $C^\infty$ Baire sense) on a closed manifold $M^{n+1}$, $3\leq (n+1)\leq 7$, we prove that the union of all closed, smooth, embedded minimal hypersurfaces is dense. This implies there are infinitely many minimal hypersurfaces thus proving a conjecture of Yau (1982) for generic metrics.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/803b66e59eed5a10ddcd05018ff19670ed355d6e

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Minimal surfaces and the Allen–Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates (2018, methodology): “s shows that these constructions can be applied to give a new proof of Yau’s conjecture on inﬁnitely many minimal surfaces in a 3-manifold with a generic metric (recently proven by Irie–Marques–Neves [IMN18]) with new geometric conclusions. Namely, we prove that a 3-manifold with a generic metric co”
Weyl law for the volume spectrum (2016, methodology): “There exists a constant a ( n ) > 0 such that, for every compact Riemannian manifold ( M n +1 , g ) with (possibly empty) boundary, we have After our paper was completed, the Weyl Law for the Volume Spectrum was used in a fundamental way in [12] to prove the following theorem: Marques, Neves, 2017).”
Existence of infinitely many minimal hypersurfaces in closed manifolds (2018, background): “Afterwards, Irie, Marques and Neves [16] settled the conjecture for generic metrics (in the Baire sense) by showing that for generic metrics on a closed manifold with the usual restriction on the dimension, the union of minimal hypersurfaces is dense.”
On the Multiplicity One Conjecture in min-max theory (2019, unknown): “For a set of generic metrics, Irie-Marques-Neves [21] and Marques-Neves- Song [30] proved denseness and equi-distribution for the space of closed embedded minimal hypersur- faces, using the Weyl Law for volume spectrum by Liokumovich-Marques-Neves [24].”
Free boundary minimal surfaces in the unit ball : recent advances and open questions (2019, methodology): “Making use of the Weyl law [68], this has been proved for generic metrics in closed Riemannian manifolds by Irie-Marques-Neves [44] and Marques-Neves-Song [72].”