Higher dimensional Sacks-Uhlenbeck-type functionals and applications
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| Format: | Preprint |
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2025
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| _version_ | 1866909654362095616 |
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| author | Di Matteo, Gianmichele Lamm, Tobias |
| author_facet | Di Matteo, Gianmichele Lamm, Tobias |
| contents | In this work, we generalize Sacks-Uhlenbeck's existence result for harmonic spheres, constructing for $n \ge 2$, regular, non-trivial, $n$-harmonic $n$-spheres into suitable target manifolds. We obtain an infinite family of new null-homotopic such maps. The proof follows a similar perturbative argument, which in high dimensions leads to a degenerate and double-phase-type Euler-Lagrange system, making the uniform regularity needed to formalize the bubbling harder to achieve. Then, we develop a refined neck-analysis leading to an energy identity along the approximation, assuming a suitable Struwe-type entropy bound along a sequence of critical points. Finally, we combine these results to solve quite general min-max problems for the $n$-energy modulo bubbling. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_17166 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Higher dimensional Sacks-Uhlenbeck-type functionals and applications Di Matteo, Gianmichele Lamm, Tobias Analysis of PDEs Differential Geometry 35-XX, 49-XX, 53-XX In this work, we generalize Sacks-Uhlenbeck's existence result for harmonic spheres, constructing for $n \ge 2$, regular, non-trivial, $n$-harmonic $n$-spheres into suitable target manifolds. We obtain an infinite family of new null-homotopic such maps. The proof follows a similar perturbative argument, which in high dimensions leads to a degenerate and double-phase-type Euler-Lagrange system, making the uniform regularity needed to formalize the bubbling harder to achieve. Then, we develop a refined neck-analysis leading to an energy identity along the approximation, assuming a suitable Struwe-type entropy bound along a sequence of critical points. Finally, we combine these results to solve quite general min-max problems for the $n$-energy modulo bubbling. |
| title | Higher dimensional Sacks-Uhlenbeck-type functionals and applications |
| topic | Analysis of PDEs Differential Geometry 35-XX, 49-XX, 53-XX |
| url | https://arxiv.org/abs/2506.17166 |