The Weyl Law for the phase transition spectrum and density of limit interfaces

Gaspar, Pedro; Guaraco, Marco A. M.

doi:10.1007/s00039-019-00489-1

Cited by 33 publications

(22 citation statements)

References 44 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(2) Our results highlight the philosophy that the solutions to Allen-Cahn provide a "canonical" approximation of the min-max surfaces. 2 We note that after the first version of this work was posted, Gaspar-Guaraco [GG19] gave a new proof of Yau's conjecture for generic metrics (in the spirit of Irie-Marques-Neves [IMN18]) by proving a Weyl law for their Allen-Cahn p-widths.…”

Section: Introductionmentioning

confidence: 99%

Minimal surfaces and the Allen--Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates

2020

View full text Add to dashboard Cite

The Allen-Cahn equation is a semilinear PDE which is deeply linked to the theory of minimal hypersurfaces via a singular limit. We prove curvature estimates and strong sheet separation estimates for stable solutions (building on recent work of Wang-Wei [WW19]) of the Allen-Cahn equation on a 3-manifold. Using these, we are able to show for generic metrics on a 3-manifold, minimal surfaces arising from Allen-Cahn solutions with bounded energy and bounded Morse index are two-sided and occur with multiplicity one and the expected Morse index. This confirms, in the Allen-Cahn setting, a strong form of the multiplicity one conjecture and the index lower bound conjecture of Marques-Neves [Mar14,Nev14] in 3-dimensions regarding min-max constructions of minimal surfaces.Allen-Cahn min-max constructions were recently carried out by Guaraco [Gua18] and Gaspar-Guaraco [GG18]. Our resolution of the multiplicity one and the index lower bound conjectures 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 contains, for every p = 1, 2, 3, . . ., a two-sided embedded minimal surface with Morse index p and area ∼ p OTIS CHODOSH AND CHRISTOS MANTOULIDIS Appendix E. An interpolation lemma 76 References 77Here, h 0 > 0 is a constant that is canonically associated with W (see Section 1.3). A deep result of Hutchinson-Tonegawa [HT00, Theorem 1] ensures that V limits to a varifold with a.e. integer density as ε ց 0. If, in addition, one assumes that the solutions are stable, Tonegawa-Wickramasekera [TW12] have shown that the limiting varifold is stable and satisfies the conditions of Wickramasekera's deep regularity theory [Wic14]; thus the limiting varifold is a smooth stable minimal hypersurface (outside of a codimension 7 singular set). In two dimensions, this was shown by Tonegawa [Ton05].1 Added in proof: There has been dramatic progress in Almgren-Pitts theory since we first posted this article. In particular, we note that A. Song [Son18] has proved the full Yau conjecture in dimensions 3 through 7, and X. Zhou [Zho19] proved the multiplicity one conjecture in the Almgren-Pitts setting, also in dimensions 3 through 7.1.2.3. The multiplicity one-conjecture for limits of the Allen-Cahn equation in 3-manifolds. In their recent work [MN16a], Marques-Neves make the following conjecture:Conjecture 1.5 (Multiplicity one conjecture). For generic metrics on (M n , g), 3 ≤ n ≤ 7, two-sided unstable components of closed minimal hypersurfaces obtained by min-max methods must have multiplicity one.In [MN16a], Marques-Neves confirm this in the case of a one parameter Almgren-Pitts sweepout. The one parameter case had been previously considered for metrics of positive Ricci curvature by Marques-Neves [MN12] and subsequently by Zhou [Zho15]. See also [Gua18, Corollary E] and [GG18, Theorem 1]for results c...

show abstract

Section: Introductionmentioning

confidence: 99%

Minimal surfaces and the Allen--Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates

2020

View full text Add to dashboard Cite

show abstract

“…(1.1) for some constant a(n) > 0. (See also [GG19].) This result has had important implications for existence of minimal hypersurfaces, cf.…”

mentioning

confidence: 89%

The p-widths of a surface

Chodosh¹,

Mantoulidis²

2021

Preprint

View full text Add to dashboard Cite

The p-widths of a closed Riemannian manifold are a nonlinear analogue of the spectrum of its Laplace-Beltrami operator, which corresponds to areas of a certain min-max sequence of minimal submanifolds. We show that the p-widths of any closed Riemannian two-manifold correspond to a union of closed immersed geodesics, rather than simply geodesic nets.We then prove optimality of the sweepouts of the round two-sphere constructed from the zero set of homogeneous polynomials, showing that the p-widths of the round sphere are attained by √ p great circles. As a result, we find the universal constant in the Liokumovich-Marques-Neves-Weyl law for surfaces to be √ π. En route to calculating the p-widths of the round two-sphere, we prove two additional new results: a bumpy metrics theorem for stationary geodesic nets, and that, generically, stationary geodesic nets with bounded mass and bounded singular set have Lusternik-Schnirelmann category zero. Contents1. Introduction 2. The p-widths and two min-max theories 3. The sine-Gordon double-well potential 4. Immersed geodesics representing the p-widths 5. The space of geodesic networks 6. Lusternik-Schnirelmann theory on a perturbed (S 2 , g 0 ) 7. The p-widths of a round 2-sphere 8. Open questions Appendix A. Metric space notions

show abstract

“…Multiparameter sweepouts in the Allen-Cahn setting were introduced by Gaspar and Guaraco [5]. In [6], Gaspar and Guaraco proved a Weyl law for the Allen-Cahn volume spectrum and were able to reproduce the proofs of density and equidistribution of minimal hypersurfaces for generic metrics. Combined with Chodosh-Mantoulidis [3], this establishes item (b) of the Morse Index Conjecture in dimension three perhaps with a different dimensional constant.…”

Section: Recent Developmentsmentioning

confidence: 99%

The space of cycles, a Weyl law for minimal hypersurfaces and Morse index estimates

Marques

Neves

2017

Surveys in Differential Geometry

View full text Add to dashboard Cite

In this note, prepared for the occasion of the Journal of Differential Geometry (JDG) 50th birthday Conference, we will discuss a Weyl law conjectured by Gromov and proved by the authors with Liokumovich in [13], and work of the authors ([19], [20]) on the characterization of the Morse index of minimal hypersurfaces produced by min-max methods.The last section is an update on dramatic developments obtained since the time of the conference. This includes the proof of Yau's Conjecture (about the existence of infinitely many minimal surfaces) for generic metrics, by establishing density of minimal hypersurfaces, obtained by the authors with Irie [11], the proof of equidistribution of minimal hypersurfaces for generic metrics by the authors with Song [21], the proof of the authors' Multiplicity One Conjecture and Morse Index Conjecture in dimension three by Chodosh and Mantoulidis ([3]), and the full resolution of Yau's Conjecture by Song [26].

show abstract

The Weyl Law for the phase transition spectrum and density of limit interfaces

Cited by 33 publications

References 44 publications

Minimal surfaces and the Allen--Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates

Minimal surfaces and the Allen--Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates

The p-widths of a surface

The space of cycles, a Weyl law for minimal hypersurfaces and Morse index estimates

Contact Info

Product

Resources

About