For each n≥2, closed targets with a non-trivial higher homotopy group admit non-trivial regular n-harmonic spheres; in dimension 3 for all metrics and in higher dimensions for homogeneous targets, with an energy identity and min-max theorem modulo bubbling.
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Higher dimensional Sacks-Uhlenbeck-type functionals and applications
For each n≥2, closed targets with a non-trivial higher homotopy group admit non-trivial regular n-harmonic spheres; in dimension 3 for all metrics and in higher dimensions for homogeneous targets, with an energy identity and min-max theorem modulo bubbling.