The minimal genus problem
This expository article gives a survey of recent progress on the the so-called minimal genus problem. The problem is the following: given a second cohomology class of a smooth four-manifold, what is the minimal genus of an embedded surface which represents the given class? The recent progress on this question has been driven mainly by the emergence of Seiberg-Witten theory. Indeed one of the very first applications of the Seiberg-Witten equations was in Kronheimer and Mrowka's proof the Thom conjecture (which gives explicit lower bound for the genus of an embedded surface representing a given cohomology class). Lawson's review focuses largely on the developments resulting from Seiberg-Witten theory, and includes background material on this important new subject.
scientific article; zbMATH DE number 1121250