Some Landmarks in the History of the Tangential Cauchy Riemann Equations

We discuss the origins of the tangential Cauchy Riemann equation beginning with W. Wirtinger in 1926, and trace the largely unknown early developments until the emergence of the ∂b−Neumann complex in the 1960s. Vienna is a most appropriate venue for a program centered on the ∂− Neumann Problem. Not only did the calculus of the differential operators ∂/∂zj and ∂/∂zj originate in the work of Wilhelm Wirtinger, Professor at the University of Vienna, but to my knowledge Wirtinger also was the first person to have thought of what today we call the tangential Cauchy Riemann equations and the corresponding notion of (tangential) Cauchy-Riemann ( = CR ) functions. Since much of the modern literature seems to be unaware of this work and of other early work on “tangential analytic functions”, it may be useful to trace the path from these origins to the modern theory of the tangential ∂−Neumann Complex as developed by J. J. Kohn and H. Rossi in the 1960s. 1. The Beginning. Wilhelm Wirtinger (1865 1945) was born in Ybbs on the Danube and studied mathematics at the Universität Wien. He earned his doctorate in 1887 with Emil Weyr and Gustav Ritter von Escherich, working on triple evolutions in the plane. For the next three years he expanded his mathematical horizons in Berlin and Göttingen, where he was strongly influenced by F. Klein. In 1890 he earned the Habilitation in Vienna, and after a few years as assistant he was appointed to a chair at the University of Innsbruck in 1895. He returned to Vienna in 1905 to assume a chair at AMS Subject Classification: Primary: 3503, 32V25; Secondary: 0102, 35N15, 35F35 This paper is based on a lecture given during the program “The ∂− Neumann Problem: Analysis, Geometry, and Potential Theory.” held at the Erwin Schrödinger International Institute for Mathematical Physics in Vienna in Fall 2009. The author gratefully acknowledges the support of the ESI during his stay in Vienna.