An index theorem for open manifolds whose ends are warped products
The author gives a careful analysis of the index theory of Dirac operators on a class of even-dimensional, complete, noncompact, spin manifolds. A manifold is in the class if it has a compact subset whose complement is topologically a cylinder and metrically a warped product (not necessarily cylindrical). The author provides some conditions sufficient to guarantee that the Dirac operator on such a manifold is Fredholm and some conditions that imply that the operator is not Fredholm. Whether or not the Dirac operator is Fredholm, it has a finite \(L^{2}\)-index, defined to be the dimension of the kernel of its restriction to positive \(L^{2}\) spinors minus the dimension of the kernel of its restriction to negative \(L^{2}\) spinors. In many cases the author provides a formula for the \(L^{2}\)-index. This formula is based on the extension of the Atiyah-Patodi-Singer index theorem due to \textit{G. Grubb} [Commun. Partial Differ. Equations 17, 2031-2077 (1992; Zbl 0773.58025)] and \textit{P. B. Gilkey} [Adv. Math. 102, 129-183 (1993; Zbl 0798.58071)]. Under some conditions the \(L^{2}\)-index is guaranteed to be zero, but in general the index formula includes two nonlocal terms.
scientific article; zbMATH DE number 1868560
scientific article