On the geometrized Skyrme and Faddeev models

TL;DR

This paper explores the mathematical properties of higher-order derivative energy functionals in Skyrme and Faddeev models, focusing on their variation formulas and stability of critical maps.

Contribution

It provides a detailed analysis of the first and second variation formulas for the $\sigma_2$-energy in these models, advancing understanding of their geometric and variational aspects.

Findings

Derived first and second variation formulas for $\sigma_2$-energy.

Identified classes of stable critical maps.

Enhanced understanding of the geometric structure of these models.

Abstract

The higher-power derivative terms involved in both Faddeev and Skyrme energy functionals correspond to $\sigma_2$-energy, introduced by Eells and Sampson. The paper provides a detailed study of the first and second variation formulae associated to this energy. Some classes of (stable) critical maps are outlined.