Minimizing energy among homotopic maps (Q1774779)

Summary: We study an energy minimizing sequence \(\{u_i\}\) in a fixed homotopy class of smooth maps from a \(3\)-manifold. After deriving an approximate monotonicity property for \(\{u_i\}\) and a continuous version of the Luckhaus lemma [\textit{L. Simon}, Theorems on regularity and singularity of energy minimizing maps. Lect. Math., ETH Zürich. Basel: Birkhäuser (1996; Zbl 0864.58015)] on \(S^2\), we show that, passing to a subsequence, \(\{u_i\}\) converges strongly in \(W^{1,2}\) topology wherever there is small energy concentration.