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.
2024
2024
De Philippis, Guido; Pigati, Alessandro
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11565/4068337
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