Abstract
Given a Hermitian line bundle Lrightarrow M over a closed, oriented Riemannian manifold M, we study the asymptotic behavior, as epsilon rightarrow 0, of couples (u_epsilon ,nabla _epsilon ) critical for the rescalings Eϵ(u,∇)=∫M(|∇u|2+ϵ2|F∇|2+14ϵ2(1-|u|2)2)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} E_\\epsilon (u,\\nabla )=\\int _M\\Big (|\\nabla u|^2+\\epsilon ^2|F_\\nabla |^2+\\frac{1}{4\\epsilon ^2}(1-|u|^2)^2\\Big ) \\end{aligned}$$\\end{document}of the self-dual Yang–Mills–Higgs energy, where u is a section of L and nabla is a Hermitian connection on L with curvature F_{nabla }. Under the natural assumption limsup _{epsilon rightarrow 0}E_epsilon (u_epsilon ,nabla _epsilon )<infty , we show that the energy measures converge subsequentially to (the weight measure mu of) a stationary integral (n-2)-varifold. Also, we show that the (n-2)-currents dual to the curvature forms converge subsequentially to 2pi Gamma , for an integral (n-2)-cycle Gamma with |Gamma |le mu . Finally, we provide a variational construction of nontrivial critical points (u_epsilon ,nabla _epsilon ) on arbitrary line bundles, satisfying a uniform energy bound. As a byproduct, we obtain a PDE proof, in codimension two, of Almgren’s existence result for (nontrivial) stationary integral (n-2)-varifolds in an arbitrary closed Riemannian manifold.
Highlights
A level set approach for the variational construction of minimal hypersurfaces was born from the work of Modica–Mortola [30], Modica [29], and Sternberg [34]
Starting from a suggestion by De Giorgi [12], they highlighted a deep connection between minimizers u : M → R of the Allen–Cahn functional
4 3 times) the perimeter functional on Several years later, Hutchinson and Tonegawa [19] initiated the asymptotic study of critical points v of F with bounded energy, without the energyminimality assumption. They showed, in particular, that their energy measures concentrate along a stationary, integral (n − 1)-varifold, given by the limit of the level sets v−1(0). These developments, together with the deep regularity work by Tonegawa and Wickramasekera on stable solutions [38], opened the doors to a fruitful min–max approach to the construction of minimal hypersurfaces, providing a PDE alternative to the rather involved discretized min–max procedure implemented by Almgren and Pitts [5,31] in the setting of geometric measure theory
Summary
A level set approach for the variational construction of minimal hypersurfaces was born from the work of Modica–Mortola [30], Modica [29], and Sternberg [34]. Hutchinson and Tonegawa [19] initiated the asymptotic study of critical points v of F with bounded energy, without the energyminimality assumption They showed, in particular, that their energy measures concentrate along a stationary, integral (n − 1)-varifold, given by the limit of the level sets v−1(0). These developments, together with the deep regularity work by Tonegawa and Wickramasekera on stable solutions [38], opened the doors to a fruitful min–max approach to the construction of minimal hypersurfaces, providing a PDE alternative to the rather involved discretized min–max procedure implemented by Almgren and Pitts [5,31] in the setting of geometric measure theory. We hope to take up this line of investigation in future work
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