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Pfaffian systems and the fundamental theorem of surface theory

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The fundamental theorem of surface theory states that there exists an immersion of a surface in three-dimensional space with prescribed first and second fundamental forms whenever the given forms satisfy the Gauß–Codazzi–Mainardi equations. The proof is based on the solution of a Pfaffian system, which is a system of first-order linear PDE , and an application of the Poincaré lemma. Consequently, the regularity of the resulting immersion crucially depends on the regularity of the solution of the corresponding Pfaffian system. This talk shall give an introduction to the classical smooth case, the existing regularity theory, and a possible extension to the optimal regularity.

This talk is part of the Cambridge Analysts' Knowledge Exchange series.

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