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

Jared Marx-Kuo

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

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arxiv 2307.09733 v2 pith:4H2Z3P37 submitted 2023-07-19 math.DG

Jared Marx-Kuo This is my paper · ORCID

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keywords energyindexinnersecondabeliancodimensioncriticalginzburg--landau

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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.

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Harmonic maps to the circle with higher dimensional singular set
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Singular S1-valued harmonic maps with prescribed codimension-2 singular set exist on closed manifolds, and three variational relaxations share the same renormalised interaction energy.

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