Minimal submanifolds from the abelian Higgs model
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Minimal submanifolds from the abelian Higgs model Alessandro Pigati1
· Daniel Stern2
Received: 1 September 2019 / Accepted: 1 September 2020 © The Author(s) 2020
Abstract Given a Hermitian line bundle L → M over a closed, oriented Riemannian manifold M, we study the asymptotic behavior, as → 0, of couples (u , ∇ ) critical for the rescalings E (u, ∇) =
M
|∇u|2 + 2 |F∇ |2 +
1 2 2 (1 − |u| ) 4 2
of the self-dual Yang–Mills–Higgs energy, where u is a section of L and ∇ is a Hermitian connection on L with curvature F∇ . Under the natural assumption lim sup→0 E (u , ∇ ) < ∞, we show that the energy measures converge subsequentially to (the weight measure μ of) a stationary integral (n − 2)varifold. Also, we show that the (n − 2)-currents dual to the curvature forms converge subsequentially to 2π , for an integral (n − 2)-cycle with || ≤ μ. Finally, we provide a variational construction of nontrivial critical points (u , ∇ ) 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.
B Alessandro Pigati
[email protected] Daniel Stern [email protected]
1
Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zurich, Switzerland
2
Department of Mathematics, Princeton University, Princeton, NJ 08544, USA
123
A. Pigati, D. Stern
1 Introduction 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 1 F (v) := |dv|2 + (1 − v 2 )2 , 4 M and two-sided minimal hypersurfaces in M, showing essentially that the functionals F -converge to ( 43 times) the perimeter functional on Caccioppoli sets. 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. This promising min–max approach based on the Allen–Cahn functionals was recently developed by Guaraco and Gaspar–Guaraco [14,16], and has been used successfully to attack some profound questions concerning the structure of min–max minimal hypersurfaces—most notably in Chodosh and Mantoulidis’s work on the multiplicity one conjecture [11]. The
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