Minimizing energy among homotopic maps

Abstract

We study an energy minimizing sequence {ui} in a fixed homotopy class of smooth maps from a 3-manifold. After deriving an approximate monotonicity property for {ui} and a continuous version of the Luckhaus lemma (Simon, 1996) on S2, we show that, passing to a subsequence, {ui} converges strongly in W1,2 topology wherever there is small energy concentration.