2025/06/20 by Di Matteo, Gianmichele, Lamm, Tobias · 1 citation
Mathematics · #35-XX #49-XX #53-XX #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2506.17166
openalex publication_date 2025/06/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this work, we generalize Sacks-Uhlenbeck's existence result for harmonic spheres, constructing for n ≥ 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.