Equidistribution of minimal hypersurfaces for generic metrics

By F. C. Marques, A. Neves, Antoine Song. Inventiones Mathematicae, 2017

122 citations recorded by the source corpus.

AbstractFor almost all Riemannian metrics (in the $$C^\infty $$C∞ Baire sense) on a closed manifold $$M^{n+1}$$Mn+1, $$3\le (n+1)\le 7$$3≤(n+1)≤7, we prove that there is a sequence of closed, smooth, embedded, connected minimal hypersurfaces that is equidistributed in M. This gives a quantitative version of the main result of Irie et al. (Ann Math 187(3):963–972, 2018), that established density of minimal hypersurfaces for generic metrics. As in Irie et al. (2018), the main tool is the Weyl Law for the Volume Spectrum proven by Liokumovich et al. (Ann Math 187(3):933–961, 2018).

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/566f5e0de9978aaba0dec6454dfb936ea047ad82

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Citations in the corpus

Morse index of multiplicity one min-max minimal hypersurfaces (2015, background): “of our Weyl Law for the Volume Spectrum ([28], with Liokumovich), which also leads to the existence of an equidistributed sequence of closed minimal hypersurfaces ([36], with Song).”
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].”
Existence of infinitely many minimal hypersurfaces in closed manifolds (2018, methodology): “This was later quantified into a generic equidistribution result in [32] by Marques, Neves and the author.”
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].”
Minimal surfaces and the Allen–Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates (2018, background): “ed) have recently shown that the set of metrics on a closed Riemannian manifold (Mn,g) (with 3 ≤ n≤ 7) with the property that the set of minimal surfaces is dense in the manifold is generic (see also [MNS19]). We note that the arguments in each of [MN17, IMN18, GG19] to prove the existence of inﬁnit”