The Allen–Cahn equation on closed manifolds

By P. Gaspar, Marco A. M. Guaraco. Calculus of Variations and Partial Differential Equations, 2016

97 citations recorded by the source corpus.

We study global variational properties of the space of solutions to -ε2Δu+W′(u)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\varepsilon ^2\Delta u + W'(u)=0$$\end{document} on any closed Riemannian manifold M. Our techniques are inspired by recent advances in the variational theory of minimal hypersurfaces and extend a well-known analogy with the theory of phase transitions. First, we show that solutions at the lowest positive energy level are either stable or obtained by min–max and have index 1. We show that if ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} is not small enough, in terms of the Cheeger constant of M, then there are no interesting solutions. However, we show that the number of min–max solutions to the equation above goes to infinity as ε→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \rightarrow 0$$\end{document} and their energies have sublinear growth. This result is sharp in the sense that for generic metrics the number of solutions is finite, for fixed ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}, as shown recently by G. Smith. We also show that the energy of the min–max solutions accumulate, as ε→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \rightarrow 0$$\end{document}, around limit-interfaces which are smooth embedded minimal hypersurfaces whose area with multiplicity grows sublinearly. For generic metrics with RicM>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{Ric}_M>0$$\end{document}, the limit-interface of the solutions at the lowest positive energy level is an embedded minimal hypersurface of least area in the sense of Mazet–Rosenberg. Finally, we prove that the min–max energy values are bounded from below by the widths of the area functional as defined by Marques–Neves.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1282fc1c7e84abed0fb4666481c2b8f2abf768f9

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

On the Multiplicity One Conjecture in min-max theory (2019, unknown): “Neves [22]. Later it was fully proved for 1-parameter case by Marques-Neves [27]. Recently, ChodoshMantoulidis [5] proved this conjecture in dimension three (n+1) = 3 for the Allen-Cahn setting; (see [12] for earlier works along this direction); they also proved that the total index is exactly kfor ”
Existence of infinitely many minimal hypersurfaces in closed manifolds (2018, methodology): “[14] M. A. M. Guaraco.”
Minimal surfaces and the Allen–Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates (2018, methodology): “e 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 thes”
Morse index of multiplicity one min-max minimal hypersurfaces (2015, methodology): “1) follows in dimension three from [9], [16] and the Weyl Law proven in Gaspar-Guaraco [17].”