Higher dimensional Sacks-Uhlenbeck approximation

In this talk, we will generalize Sacks-Uhlenbeck’s existence of harmonic 2-spheres result, to higher dimensional domains, that is we find non-trivial n-harmonic n-spheres in suitable targets. Due to the conformal invariance of the n-energy, we follow a similar perturbation method, which however leads to a degenerate Euler-Lagrange system in high dimensions, making the regularity theory harder to achieve. Then we formalize the bubbling phenomenon along a sequence of critical maps and, through a refined neck-analysis, we obtain a quantization of the energy result, under a suitable Struwe-type entropy bound on the sequence.