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

2022/05/24 by Guido De Philippis, De Philippis, Guido, Alessandro Pigati +1 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2205.12389

openalex publication_date 2022/05/24 · openalex created_date 2022/06/13 · openalex updated_date 2026/07/28

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.

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