Calculus of Variations and Geometric Measure Theory

G. De Philippis - A. Pigati

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

created by dephilipp on 05 Jun 2023



Inserted: 5 jun 2023

Year: 2022

ArXiv: 2205.12389 PDF


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.