Weyl law for the volume spectrum

By Yevgeny Liokumovich, F. C. Marques, A. Neves. 2016

145 citations recorded by the source corpus.

Given $M$ a Riemannian manifold with (possibly empty) boundary, we show that its volume spectrum $\{\omega_p(M)\}_{p\in\mathbb{N}}$ satisfies a Weyl law that was conjectured by Gromov.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7dc35fd7150c8c9e2abd3bf26c04492412a4c052

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

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].”
Existence of infinitely many minimal hypersurfaces in positive Ricci curvature (2013, background): “The authors and Liokumovich confirmed this conjecture in [25].”
Density of minimal hypersurfaces for generic metrics. (2017, methodology): “The main result of [10] used in this paper is Weyl Law for the Volume Spectrum (Liokumovich, Marques, Neves, 2016).”
Existence of infinitely many minimal hypersurfaces in closed manifolds (2018, methodology): “The second formula in (5) follows from a Lusternick-Schnirelman type argument in the setting of Almgren-Pitts theory already used by Marques-Neves [26] and then by Liokumovich-Marques-Neves [22].”
Min-max theory for free boundary minimal hypersurfaces, I: Regularity theory (2016, background): “jecture [1] and Yau’s conjecture [31] in the case of positive Ricci curvature. Due to these tremendous success, there have been a vast number of exciting developments on this subject (see for example [26] [30] [45] [46]). For the constrained free boundary problem, it was claimed in [23, Theorem 4.1]”
Morse index of multiplicity one min-max minimal hypersurfaces (2015, methodology): “1 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).”
Minimal surfaces and the Allen–Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates (2018, methodology): “ (Mn,g) (for 3 ≤ n≤ 7) with positive Ricci curvature admits inﬁnitely many minimal surfaces. Moreover, by an ingenious application of the Weyl law for the p-widths proven by Liokumovich–Marques–Neves [LMN18], Irie–Marques–Neves [IMN18] (see also the recent work of Gaspar–Guaraco [GG19] that appeared”