On energy gap of unitons (Q1849434)

In [Commun. Anal. Geom. 3, 371-419 (1995; Zbl 0848.58013) and J. Math. Soc. Japan 50, 737-751 (1998; Zbl 0943.58010)], \textit{C. K. Anand} showed that unitons are equivalent to holomorphic ``uniton bundles'', with energy corresponding to the bundles' second Chern class. Using monad representation, Anand obtained a simple formula for the unitons and proved that 2-unitons have normalized energy at least four. This bound is sharp. In a personal communication, Anand proposed the following conjecture: \(m\)-unitons have energy at least \(m^2\). The paper under review gives a positive answer to this conjecture: Let \(\phi: S^2 \to U(N)\) be a harmonic map with minimal uniton number \(m\). Then \[ E(\phi) \geq 4\pi m^2. \] The author also decides the cases when equality holds. The proof is based on the work of K. Uhlenbeck, the key idea is to consider the extended harmonic map and deform it continuously to a simpler one. These procedures can only decrease the energy, and then the computation about the energy of the resulting map gives the desired estimate.