Morse Index Stability for the Ginzburg-Landau Approximation

Abstract

In this paper we study the behaviour of critical points of the Ginzburg-Landau perturbation of the Dirichlet energy into the sphere $E_{\epsilon }(u):={\int }_{\Sigma }\frac{1}{2}|du{|}_h^{2}dvo{l}_h+\frac{1}{4{\epsilon }^2}(1-|u{|}^2{)}^{2}dvo{l}_h={\int }_{\Sigma }{e}_{\epsilon }(u)$. Our first main result is a precise point-wise estimate for $e_{\epsilon }(u_k)$ in the regions where compactness fails, which also implies the $L^{2,1}$ quantization in the bubbling process. Our second main result consists in applying the method developed in a previous joint paper with T. Rivière to study the upper-semi-continuity of the extended Morse index to sequences of critical points of $E_{\epsilon }$: given a sequence of critical points $u_{{\epsilon }_k}:\Sigma \to {\mathbb{R}}^{n+1}$ of $E_{\epsilon }$ that converges in the bubble tree sense to a harmonic map $u_{\infty }\in W^{1,2}(\Sigma ,S^n)$ and bubbles $v_{\infty }^i:{\mathbb{R}}^2\to S^n$, we show that the extended Morse indices of the maps $v^i,u_{\infty }$ control the extended Morse index of the sequence $u_{{\epsilon }_k}$ for $k$ large enough.

30 pages, internal references were broken in v1