Equivalence problem for Bishop surfaces

The paper has two parts. We first briefly survey recent studies on the equivalence problem for real submanifolds in a complex space under the action of biholomorphic transformations. We will mainly focus on some of the recent studies of Bishop surfaces, which, in particular, includes the work of the authors. In the second part of the paper, we apply the general theory developed by the authors to explicitly classify an algebraic family of Bishop surfaces with a vanishing Bishop invariant. More precisely, we ...