Second Inner Variations of Energy and Index of Codimension $2$ Minimal Submanifolds

TL;DR

This paper computes the second inner variation of certain energies and relates the Morse index of limiting minimal submanifolds to the index of critical points, advancing understanding of variational stability in geometric analysis.

Contribution

It introduces a method to bound the Morse index of codimension 2 minimal submanifolds using second inner variation formulas and energy measure convergence.

Findings

Bound on Morse index of minimal submanifolds

Convergence of energy measures and stress-energy tensors

Relation between critical points and minimal submanifolds

Abstract

We compute the second inner variation of the Abelian Yang--Mills--Higgs and Ginzburg--Landau energies. Given a sequence of critical points with energy measures converging to a codimension $2$ minimal submanifold, we use the second inner variation formula to bound the morse index of the submanifold by the index of the critical points. The key tools are the convergence of the energy measures and the stress-energy tensors of solutions to Abelian Yang--Mills--Higgs and Ginzburg--Landau equations.