Given a Hermitian line bundle L over a closed, oriented Riemannian manifold M, we study the asymptotic behavior, as a scaling parameter goes to zero, of couples (section, connection) critical for the rescalings of the self-dual Yang-Mills-Higgs energy. Under a natural energy growth assumption, we show that the energy measures converge subsequentially to (the weight measure of) a stationary integral varifold. Also, we show that the currents dual to the curvature forms converge subsequentially to (a multiple of) an integral cycle pointwise bounded by the varifold. Finally, we provide a variational construction of nontrivial critical points 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 varifolds in an arbitrary closed Riemannian manifold.
Minimal submanifolds from the abelian Higgs model
Pigati, Alessandro;
2021
Abstract
Given a Hermitian line bundle L over a closed, oriented Riemannian manifold M, we study the asymptotic behavior, as a scaling parameter goes to zero, of couples (section, connection) critical for the rescalings of the self-dual Yang-Mills-Higgs energy. Under a natural energy growth assumption, we show that the energy measures converge subsequentially to (the weight measure of) a stationary integral varifold. Also, we show that the currents dual to the curvature forms converge subsequentially to (a multiple of) an integral cycle pointwise bounded by the varifold. Finally, we provide a variational construction of nontrivial critical points 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 varifolds in an arbitrary closed Riemannian manifold.
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