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.
2021
2020
Pigati, Alessandro; Stern, Daniel
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11565/4061141
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