Morse Index Stability for the Ginzburg-Landau Approximation
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909223509557248 |
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| author | Da Lio, Francesca Gianocca, Matilde |
| author_facet | Da Lio, Francesca Gianocca, Matilde |
| contents | In this paper we study the behaviour of critical points of the Ginzburg-Landau perturbation of the Dirichlet energy into the sphere $E_\varepsilon(u):=\int_Σ\frac{1}{2}|du|^2_h\ \,dvol_h +\frac{1}{4\varepsilon^2}(1-|u|^2)^2\,dvol_h=\int_Σe_{\varepsilon}(u)$. 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. Rivière to study the upper-semi-continuity of the extended Morse index to sequences of critical points of $E_ε$: given a sequence of critical points $u_{\varepsilon_k}:Σ\to \mathbb{R}^{n+1}$ of $E_\varepsilon$ that converges in the bubble tree sense to a harmonic map $u_\infty\in W^{1,2}(Σ,{S}^{n})$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_07317 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Morse Index Stability for the Ginzburg-Landau Approximation Da Lio, Francesca Gianocca, Matilde Differential Geometry Analysis of PDEs 53C43, 58E20, 58J05, 58E05, 35A15, 35J20 In this paper we study the behaviour of critical points of the Ginzburg-Landau perturbation of the Dirichlet energy into the sphere $E_\varepsilon(u):=\int_Σ\frac{1}{2}|du|^2_h\ \,dvol_h +\frac{1}{4\varepsilon^2}(1-|u|^2)^2\,dvol_h=\int_Σe_{\varepsilon}(u)$. 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. Rivière to study the upper-semi-continuity of the extended Morse index to sequences of critical points of $E_ε$: given a sequence of critical points $u_{\varepsilon_k}:Σ\to \mathbb{R}^{n+1}$ of $E_\varepsilon$ that converges in the bubble tree sense to a harmonic map $u_\infty\in W^{1,2}(Σ,{S}^{n})$ 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. |
| title | Morse Index Stability for the Ginzburg-Landau Approximation |
| topic | Differential Geometry Analysis of PDEs 53C43, 58E20, 58J05, 58E05, 35A15, 35J20 |
| url | https://arxiv.org/abs/2406.07317 |