Solutions of the Ginzburg-Landau equations concentrating on codimension-2 minimal submanifolds

Abstract

We consider the magnetic Ginzburg-Landau equations in a compact manifold Nformally corresponding to the Euler-Lagrange equations for the energy functionalHere u : N → C and A is a 1-form on N . Given a codimension-2 minimal submanifold M ⊂ N which is also oriented and non-degenerate, we construct a solution (uε, Aε) such that uε has a zero set consisting of a smooth surface close to M . Away from M we haveas ε → 0, for all sufficiently small z = 0 and y ∈ M . Here, {ν 1 , ν 2 } is a normal frame for M in N . This improves a recent result by De Philippis and Pigati [10] who built a solution for which the concentration phenomenon holds in an energy, measure-theoretical sense.We consider a closed, n-dimensional manifold N n and a closed n − 2-dimensional minimal submanifold M n−2 ⊂ N n . We say that a minimal manifold. We also require that M is non-degenerate, in the sense that the Jacobi operator has trivial bounded kernel, namely(1.6)We recall that the Jacobi operator is the second variation of the area functional around M , namelyNow we state the main result.