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arXiv · 2506.17166

Higher dimensional Sacks-Uhlenbeck-type functionals and applications

Gianmichele Di Matteo·Tobias Lamm
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Abstract

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.

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Gianmichele Di Matteo, Tobias Lamm. 2025-06-20. Higher dimensional Sacks-Uhlenbeck-type functionals and applications. https://arxiv.org/abs/2506.17166

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