Asymptotics of Minimizers for Ginzburg--Landau-type Functionals in High Dimensions

TL;DR

This paper analyzes the asymptotic behavior of local minimizers for Ginzburg--Landau-type functionals in high dimensions, revealing convergence to rectifiable measures and harmonic maps with controlled singularities.

Contribution

It establishes the convergence of energy measures to rectifiable varifolds and describes the regularity and singularity structure of minimizers in high dimensions.

Findings

Normalized energy measures converge to an $(n-2)$-rectifiable measure.

Minimizers converge strongly to harmonic maps outside a small singular set.

Quantized density relates to the homotopy classes of the vacuum manifold.

Abstract

We investigate local minimizers of Ginzburg--Landau-type functionals in dimension $n\geq 3$ that satisfy logarithmic energy bounds, assuming the potential has a vacuum manifold with a finite fundamental group. We show that the normalized energy measures converge to an $(n-2)$-rectifiable measure associated with a stationary varifold, with quantized density determined by the homotopy classes of the vacuum manifold. Away from the support of the $(n-2)$-rectifiable measure, the minimizers converge strongly in $H^1_{\text{loc}}$ to a minimizing harmonic map, which is smooth outside an $(n-3)$-rectifiable singular set.