Energy identity for a higher dimensional Sacks-Uhlenbeck approximation

In this talk, we introduce a family of functionals approximating the conformally invariant Dirichlet n-energy of maps between two Riemannian manifolds (M^n,g) and (N,h), which admit critical points. Along the approximation process, these critical points may incur a bubbling phenomenon, due to the conformal invariance of the limit Dirichlet n-energy. We prove an energy identity result for this approximation, ensuring that no energy gets lost along the formation of bubbles, under a Struwe type entropy bound assumption. We then show that min-max problems for the n-energy are always solved by a "bubble tree" of n-harmonic maps. This is a joint work with T. Lamm