Complete, embedded, minimal n‐dimensional submanifolds in ℂⁿ

Special Lagrangian conifolds, II: gluing constructions in ℂ m

“…Example 6.5. This result extends the previous work by Arezzo-Pacard [1] which had produced minimal (but not Lagrangian) desingularizations of similar configurations of SL planes under additional technical hypotheses. In particular, our construction produces the first examples of smooth SL conifolds in C m with three or more planar ends.…”

“…There exists a function C = C(x) > 0, compactly supported in a neighbourhood of the singularities, such that, for all t as in (1), all (ẽ, v, f ) ∈B t α and all x in the ith component ofL,…”

“…The fact that (L t , ι t ) is SL away from the necks implies that t ι * t (ReΩ) ≡ 1 away from the necks, so P t = Δ gt there. The proofs of Corollary 2.13 and Theorem 4.8 depend mostly on the properties of the operator listed in (1). The only additional ingredient is the rescaling property Δ t 2 g = t −2 Δ g .…”

Entire solutions of the magnetic Ginzburg-Landau equation in $\mathbb{R}^4$

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We construct entire solutions of the magnetic Ginzburg-Landau equations in dimension 4 using Lyapunov-Schmidt reduction. The zero set of these solutions are close to the minimal submanifolds studied by Arezzo-Pacard [1]. We also show the existence of a saddle type solution to the equations, whose zero set consists of two vertical planes in R 4 .

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“…To proceed, let us recall the following minimal submanifold studied in Arezzo-Pacard [1]. Let Γ = {(ρe iθ , : ρ −1 e iθ ) : ρ, θ ∈ R}.…”