Higher dimensional Sacks-Uhlenbeck approximation
Relatore
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Gianmichele Di MatteoScuola Normale Superiore
Gianmichele Di Matteo - Scuola Normale Superiore
Higher dimensional Sacks-Uhlenbeck approximation
Abstract
In this talk, we will introduce a generalization of Sacks-Uhlenbeck’s existence of harmonic 2-spheres result to higher dimensional domains, that is we construct non-trivial, regular, n-harmonic n-spheres in suitable target manifolds. 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 formalise the bubbling harder to achieve. Then, we develop a refined neck-analysis leading to a quantization of the energy 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.