Morse Index Stability for the Ginzburg-Landau Approximation

In this paper we study the behaviour of critical points of the Ginzburg-Landau perturbation of the Dirichlet energy into the sphere E_ε(u):=∫_Σ 1/2|du|²ₕ\ \,dvolₕ +1/4ε²(1-|u|²)²\,dvolₕ=∫Σeε(u). Our first main result is a precise point-wise estimate for e_ε(uₖ) in the regions where compactness fails, which also implies the L2,1 quantization in the bubbling process. Our second main result consists in applying the method developed in a previous joint paper with T. Rivi\`ere to study the upper-semi-continuity of the extended Morse index to sequences of critical points of Eε: given a sequence of critical points uεₖ:Σ→ Rⁿ⁺¹ of E_ε that converges in the bubble tree sense to a harmonic map u_∞∈ W1,2(Σ,Sⁿ) and bubbles vⁱ∞:R²→ Sⁿ, we show that the extended Morse indices of the maps vⁱ,u_∞ control the extended Morse index of the sequence uεₖ for k large enough.