Existence of infinitely many homotopy classes from $\mathbb S^3$ to $\mathbb S^2$ having a minimimzing $W^{s,\frac 3s}$-harmonic map
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| Format: | Preprint |
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2025
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| _version_ | 1866913878143664128 |
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| author | Grzela, Adam Mazowiecka, Katarzyna |
| author_facet | Grzela, Adam Mazowiecka, Katarzyna |
| contents | In 1998 T. Rivière proved that there exist infinitely many homotopy classes of $π_3(\mathbb S^2)$ having a minimizing 3-harmonic map. This result is especially surprising taking into account that in $π_3(\mathbb S^3)$ there are only three homotopy classes (corresponding to the degrees $\{-1,0,1\}$) in which a minimizer exists.
We extend this theorem in the framework of fractional harmonic maps and prove that for $s\in(0,1)$ there exist infinitely many homotopy classes of $π_{3}(\mathbb S^{2})$ in which there is a minimizing $W^{s,\frac{3}{s}}$-harmonic map. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_05060 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence of infinitely many homotopy classes from $\mathbb S^3$ to $\mathbb S^2$ having a minimimzing $W^{s,\frac 3s}$-harmonic map Grzela, Adam Mazowiecka, Katarzyna Analysis of PDEs In 1998 T. Rivière proved that there exist infinitely many homotopy classes of $π_3(\mathbb S^2)$ having a minimizing 3-harmonic map. This result is especially surprising taking into account that in $π_3(\mathbb S^3)$ there are only three homotopy classes (corresponding to the degrees $\{-1,0,1\}$) in which a minimizer exists. We extend this theorem in the framework of fractional harmonic maps and prove that for $s\in(0,1)$ there exist infinitely many homotopy classes of $π_{3}(\mathbb S^{2})$ in which there is a minimizing $W^{s,\frac{3}{s}}$-harmonic map. |
| title | Existence of infinitely many homotopy classes from $\mathbb S^3$ to $\mathbb S^2$ having a minimimzing $W^{s,\frac 3s}$-harmonic map |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2506.05060 |