The $\bar \partial$-Problem

Abstract

Differential forms are often introduced purely algebraically (Spivak [1], Gunning-Rossi [1]), as members of a graded ring, or simply as “formal sums” that are to be manipulated according to the rules of “exterior algebra,” but they can also be defined as complex-valued functions whose domain is the collection of all suitably differentiable surfaces of the appropriate dimension. We shall sketch this second approach, omitting all proofs; they are elementary, but long-winded and repetitious. Details may be found in Rudin [16]. The main purpose of this introductory section is to recall the basic facts and to establish notation.