Celebratio Mathematica — Uhlenbeck

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Works connected to Shiing-Shen Chern

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K. Uhlenbeck: “Regularity theorems for solutions of elliptic polynomial equations,” pp. 225–231 in Global analysis (Berkeley, CA, 1–26 July 1968). Edited by S.-S. Chern and S. Smale. Proceedings of Symposia in Pure Mathematics 16. American Mathematical Society (Providence), 1970. MR 413168 Zbl 0216.38202 incollection

Stephen Smale	Related
S. S. Chern	Related	Volume

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