تقرير
Morse Index Stability for the Ginzburg-Landau Approximation
| العنوان: | Morse Index Stability for the Ginzburg-Landau Approximation |
|---|---|
| المؤلفون: | Da Lio, Francesca, Gianocca, Matilde |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Differential Geometry, Mathematics - Analysis of PDEs, 53C43, 58E20, 58J05, 58E05, 35A15, 35J20 |
| الوصف: | 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. Comment: 30 pages, internal references were broken in v1 |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2406.07317 |
| رقم الأكسشن: | edsarx.2406.07317 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |