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. While for the latter energies gluing methods are also effective, in general dimension our proof is by now the only available one in the Ginzburg-Landau setting, where the weaker energy concentration is the main technical difficulty.
Non‐degenerate minimal submanifolds as energy concentration sets: a variational approach
Pigati, Alessandro
2024
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. While for the latter energies gluing methods are also effective, in general dimension our proof is by now the only available one in the Ginzburg-Landau setting, where the weaker energy concentration is the main technical difficulty.File | Dimensione | Formato | |
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