Non-degenerate minimal submanifolds as energy concentration sets: a variational approach

TL;DR

This paper demonstrates that non-degenerate minimal submanifolds of codimension two can be characterized as energy concentration sets of critical maps for certain variational energies, extending previous results to multiple models.

Contribution

It provides a purely variational proof linking minimal submanifolds to energy concentration sets across several important energies.

Findings

Non-degenerate minimal submanifolds are energy concentration sets for Ginzburg-Landau maps.

The proof extends to U(1)-Yang-Mills-Higgs and Allen-Cahn-Hilliard energies.

The approach generalizes previous geodesic results to higher codimension submanifolds.

Abstract

We prove that every non-degenerate minimal submanifold of codimension two can be obtained as the energy concentration set of a family of critical maps for the (rescaled) Ginzburg-Landau functional. The proof is purely variational, and follows the strategy laid out by Jerrard and Sternberg, extending a recent result for geodesics by Colinet-Jerrard-Sternberg. The same proof applies also to the $U(1)$-Yang-Mills-Higgs and to the Allen-Cahn-Hilliard energies.