On the Index of Elliptic Operators on Closed Surfaces

Abstract

The index of a linear mapping is the difference between the dimension of its null space and the codimension of its range. Many results concerning the index of linear elliptic operators on two-dimensional domains have been known for some time. In a recent article by I. M. Gelfand [12], a systematic study of the index of general elliptic problems as a homotopy invariant was suggested. Subsequently articles by Volpert, Dynin, and Agranovic [1, 2, 8, 9, 23, 24] have contributed substantially to the understanding of the nature of this problem. The author has recently learned that M. F. Atiyah and 1. M. Singer have obtained a formula for the index of very general elliptic problems on compact manifolds of any dimension. Unfortunately, these resuLlts are unavailable to him at this time.2 The present paper was inspired by two announcements of Volpert [23, 24], concerning elliptic problems on the sphere, S2. The second of these contains an especially simple formula for the index in terms of the degree of a related mapping. In this work, we show that this formula remains valid for any closed oriented surface. Furthermore, we make a detailed study of the structure of elliptic operators of first order on such a surface. For the convenience of the reader, we have included an exposition of known results in ?? 1-3. With slight modifications, these remain valid for compact mnanifolds of arbitrary finite dimension. In ? 4, we define a homomorphism K, which sends the group of continuous mappings of the cotangent sphere bundle B of the surface into the general linear group GL (n, C), into the additive group of integers. The value of K on a mapping a: 3 -> GL(n, C) of class C? is just the index of a system of singular integral operators, whose sym-bol is the fuclition C. ? 5 is devoted to a study of the general structure of elliptic operators of first order on closed surfaces. In ? 6, we define the degree 1(C) of a mapping CX: B -> GL(n, C). This is just the degree of the