Morse Index Stability for the Ginzburg-Landau Approximation

TL;DR

This paper investigates the stability of critical points of the Ginzburg-Landau energy, providing point-wise estimates and analyzing the semi-continuity of the Morse index during bubbling and convergence to harmonic maps.

Contribution

It introduces precise point-wise estimates for energy density in regions of non-compactness and extends Morse index semi-continuity results to critical points converging to harmonic maps.

Findings

Established $L^{2,1}$ quantization in bubbling regions.

Proved upper semi-continuity of Morse index during bubble convergence.

Linked Morse indices of bubbles and limit maps to the original sequence.

Abstract

In this paper we study the behaviour of critical points of the Ginzburg-Landau perturbation of the Dirichlet energy into the sphere $E_\varepsilon(u):=\int_\Sigma \frac{1}{2}|du|^2_h\ \,dvol_h +\frac{1}{4\varepsilon^2}(1-|u|^2)^2\,dvol_h=\int_{\Sigma}e_{\varepsilon}(u)$. Our first main result is a precise point-wise estimate for $e_\varepsilon(u_k)$ 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. Rivi\`ere 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 $u_{\varepsilon_k}:\Sigma\to \mathbb{R}^{n+1}$ of $E_\varepsilon$ that converges in the bubble tree sense to a harmonic map $u_\infty\in W^{1,2}(\Sigma,{S}^{n})$ and bubbles $v^i_{\infty}:\mathbb{R}^2\to {S}^{n}$, we show that the extended Morse indices of the maps $v^i,u_\infty$ control the extended Morse index of the sequence $u_{\varepsilon_k}$ for $k$ large enough.