Entire solutions to 4-dimensional Ginzburg-Landau equations and codimension 2 minimal submanifolds

Abstract

We consider the magnetic Ginzburg-Landau equations informally corresponding to the Euler-Lagrange equations for the energy functionalHere u : R 4 → C, A : R 4 → R 4 and d denotes exterior derivative when A is regarded as a 1-form in R 4 . Given a 2-dimensional minimal surface M in R 3 with finite total curvature and non-degenerate, we construct a solution (uε, Aε) which has a zero set consisting of a smooth 2-dimensional surface close to M × {0} ⊂ R 4 . Away from the latter surface we have |uε| → 1 andfor all sufficiently small z = 0. Here y ∈ M and ν(y) is a unit normal vector field to M in R 3 .

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

“…This result has already been established in [2,Proposition 4]. We sketch the proof for completeness.…”

“…Our strategy relies on the construction of of an accurate approximation to a solution, the use of linearization and the so-called outer-inner gluing method to set up a suitable fixed point formulation. We have recently used this approach in a similar context in R 4 obtaining concentration for a special class of non-compact minimal surfaces embedded in R 3 which includes a catenoid and the Costa-Hoffman-Meeks minimal surfaces [2]. Liu, Ma, Wei and Wu [19] have obtained the existence of a solution in R 4 with precise asymptotics and concentration on a special non-compact codimension-2 symmetric minimal manifolds discovered by Arezzo and Pacard [1].…”

“…The role of gauge invariance in the linearized equation. Here we recall briefly the content of §2.2 of [2]. To prove Theorem 1 we first construct an approximate solution W , whose profile close to the manifold resembles the behaviour of U 0 , then we look for true solutions as perturbations Φ of the first approximation W , namely we solve S ε (W + Φ) = 0.…”