Yang-Mills-Higgs fields and harmonicity of limit maps (Q911115)

This is a preliminary announcement of a result that asserts harmonicity of the limit map \(\Phi_{\infty}\) of the Higgs field at infinity corresponding to a finite SU(2) Yang-Mills-Higgs field (A,\(\Phi)\) provided that \(\lim_{R\to \infty}\sup_{| x| \leq R}| | \Phi | (x)-1| =0\) and \(\lim_{R\to \infty}<R^ 2A(Rx),\Phi (Rx)>=0\) for all \(x\in S^ 2\). A generalization is stated replacing SU(2) by an arbitrary compact simple Lie group G. Finally, under the Donaldson correspondence, a similar argument links magnetic monopoles and holomorphic maps.