Abstract
We prove a Weyl Law for the phase transition spectrum based on the techniques of Liokumovich–Marques–Neves. As an application we give phase transition adaptations of the proofs of the density and equidistribution of minimal hypersufaces for generic metrics by Irie–Marques–Neves and Marques–Neves–Song, respectively. We also prove the density of separating limit interfaces for generic metrics in dimension 3, based on the recent work of Chodosh–Mantoulidis, and for generic metrics on manifolds containing only separating minimal hypersurfaces, e.g. \({H_{n}(M,\mathbb{Z}_2) = 0}\), for \({4 \leq n + 1 \leq 7}\). These provide alternative proofs of Yau’s conjecture on the existence of infinitely many minimal hypersurfaces for generic metrics on each setting, using the Allen–Cahn approach.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ambrosio, L., Cabré, X.: Entire solutions of semilinear elliptic equations in \(\mathbb{R}^3\) and a conjecture of De Giorgi. J. Am. Math. Soc. 13, 725–739 (2000)
Cahn, J., Allen, S.: A microscopic theory for domain wall motion and its experimental verification in Fe-Al alloy domain growth kinetics. Le Journal de Physique Colloques 38, C7–51 (1977)
O. Chodosh and C. Mantoulidis. Minimal surfaces and the Allen–Cahn equation on 3-manifolds: Index, multiplicity, and curvature estimates. arXiv:1803.02716 [math.DG], (2018)
Fadell, E.R., Rabinowitz, P.H.: Bifurcation for odd potential operators and an alternative topological index. Journal of Functional Analysis 26, 48–67 (1977)
P. Gaspar. The second inner variation of energy and the Morse index of limit interfaces. arXiv:1710.04719 [math.DG], (2017)
Gaspar, P., Guaraco, M.A.: The Allen-Cahn equation on closed manifolds. Calc. Var. Partial Differ. Equ. 57, 101 (2018)
Ghoussoub, N.: Location, multiplicity and Morse indices of min-max critical points. J. Reine Angew. Math. 417, 27–76 (1991)
M. Gromov. Dimension, nonlinear spectra and width, geometric aspects of functional analysis (1986/87), 132–184, Lecture Notes in Math, 1317 (1986/87)
Gromov, M.: Isoperimetry of waists and concentration of maps. Geom. Funct. Anal. 13, 178–215 (2003)
M. Gromov. Singularities, expanders and topology of maps. I. Homology versus volume in the spaces of cycles. Geom. Funct. Anal., 19 (2009), 743–841
Guaraco, M.A.M.: Min-max for phase transitions and the existence of embedded minimal hypersurfaces. J. Differ. Geom. 108, 91–133 (2018)
Guth, L.: The width-volume inequality. Geom. Funct. Anal. 17, 1139–1179 (2007)
Guth, L.: Minimax problems related to cup powers and Steenrod squares. Geom. Funct. Anal. 18, 1917–1987 (2009)
F. Hiesmayr. Spectrum and index of two-sided Allen–Cahn minimal hypersurfaces. arXiv:1704.07738 [math.DG], (2017)
Hutchinson, J.E., Tonegawa, Y.: Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory. Calc. Var. Partial Differ. Equ. 10, 49–84 (2000)
K. Irie. F. Marques, and A. Neves, Density of minimal hypersurfaces for generic metrics. Ann. of Math. (2), 187 (2018), 963–972
Kohn, R.V., Sternberg, P.: Local minimisers and singular perturbations. P. R. Soc. Edinb. A. 111, 69–84 (1989)
Y. Liokumovich, F. Marques, and A. Neves. Weyl law for the volume spectrum. Ann. of Math. (2), 187 (2018), 933–961
C. Mantoulidis. Allen–Cahn min-max on surfaces. arXiv:1706.05946 [math.AP], (2017)
Marques, F.C., Neves, A.: Morse index and multiplicity of min-max minimal hypersurfaces. Camb. J. Math. 4, 463–511 (2016)
Marques, F.C., Neves, A.: Existence of infinitely many minimal hypersurfaces in positive Ricci curvature. Invent. Math. 209, 577–616 (2017)
F. C. Marques, A. Neves, and A. Song. Equidistribution of minimal hypersurfaces for generic metrics. arXiv:1712.06238 [math.DG], (2017)
Mazet, L., Rosenberg, H.: Minimal hypersurfaces of least area. J. Differ. Geom. 106, 283–316 (2017)
Modica, L.: The gradient theory of phase transitions and the minimal interface criterion. Arch. Ration. Mech. Anal. 98, 123–142 (1987)
L. Modica and S. Mortola. Il limite nella \(\Gamma \)-convergenza di una famiglia di funzionali ellittici. Boll. Un. Mat. Ital. A (5), 14 (1977), 526–529
F. Pacard. The role of minimal surfaces in the study of the Allen–Cahn equation, in Geometric analysis: Partial differential equations and surfaces, vol. 570 of Contemp. Math., Amer. Math. Soc., Providence, RI, (2012), pp. 137–163
Pacard, F., Ritoré, M.: From constant mean curvature hypersurfaces to the gradient theory of phase transitions. J. Differ. Geom. 64, 359–423 (2003)
Savin, O.: Phase transitions, minimal surfaces and a conjecture of de Giorgi. Curr. Dev. Math. 2009, 59–114 (2009)
Sharp, B.: Compactness of minimal hypersurfaces with bounded index. J. Differ. Geom. 106, 317–339 (2017)
S. Smale. An infinite dimensional version of Sard's theorem, in The Collected Papers of Stephen Smale: Volume 2, World Scientific, (2000), pp. 529–534
Sternberg, P.: The effect of a singular perturbation on nonconvex variational problems. Arch. Ration. Mech. Anal. 101, 209–260 (1988)
Tonegawa, Y.: On stable critical points for a singular perturbation problem. Commun. Anal. Geom. 13, 439–459 (2005)
Tonegawa, Y., Wickramasekera, N.: Stable phase interfaces in the van der Waals-Cahn-Hilliard theory. J. Reine Angew. Math. 668, 191–210 (2012)
Vannella, G.: Existence and multiplicity of solutions for a nonlinear Neumann problem. Ann. Mat. Pura Appl. 180, 429–440 (2002)
K. Wang and J. Wei. Finite morse index implies finite ends. arXiv:1705.06831 [math.AP], (2017)
B. White. The space of minimal submanifolds for varying riemannian metrics. Indiana Univ. Math. J., (1991), 161–200
White, B.: On the bumpy metrics theorem for minimal submanifolds. Am. J. Math. 139, 1149–1155 (2017)
N. Wickramasekera. A general regularity theory for stable codimension 1 integral varifolds. Ann. of Math. (2), 179 (2014), 843–1007
Acknowledgements
Both authors would like to thank Fernando C. Marques and André Neves for useful discussions and their interest in this work. The first author is grateful to the Department of Mathematics at Princeton University for its hospitality. Part of this work and the first drafts were carried out while visiting during the academic year of 2017–2018. The second author would like to thank FIM - ETH, Zurich for their kind hospitality, where this work was finished during a visit in Spring 2018.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The first author was partly supported by NSF Grant DMS-1311795.
Rights and permissions
About this article
Cite this article
Gaspar, P., Guaraco, M.A.M. The Weyl Law for the phase transition spectrum and density of limit interfaces. Geom. Funct. Anal. 29, 382–410 (2019). https://doi.org/10.1007/s00039-019-00489-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00039-019-00489-1