Celebratio Mathematica — Uhlenbeck

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B. Dai, C.-L. Terng, and K. Uhlenbeck: “On the space-time monopole equation,” pp. 1–30 in Essays in geometry in memory of S. S. Chern. Edited by S.-T. Yau. Surveys in Differential Geometry 10. International Press (Somerville, MA), 2006. MR 2408220 Zbl 1157.53016 incollection

The space-time monopole equation is obtained from a dimension reduction of the anti-self dual Yang–Mills equation on \( \mathbb{R}^{2,2} \). A family of Ward equations is obtained by gauge fixing from the monopole equation. In this paper, we give an introduction and a survey of the space-time monopole equation. Included are alternative explanations of results of Ward, Fokas–Ioannidou, Villarroel and Zakhorov–Mikhailov. The equations are formulated in terms of a number of equivalent Lax pairs; we make use of the natural Lorentz action on the Lax pairs and frames. A new Hamiltonian formulation for the Ward equations is introduced. We outline both scattering and inverse scattering theory and use Bäcklund transformations to construct a large class of monopoles which are global in time and have both continuous and discrete scattering data.

Chuu-Lian Terng	Related
S. S. Chern	Related	Volume
Shing-Tung Yau	Related
Bo Dai	Related