On index formulas for manifolds with metric horns

Abstract

In this paper we discuss the index problem for geometric differential operators (Spin-Dirac operator, Gau{\ss}-Bonnet operator, Signature operator) on manifolds with metric horns. On singular manifolds these operators in general do not have unique closed extensions. But there always exist two extremal extensions $D_{min}$ and $D_{max}$. We describe the quotient ${\cal D}(D_{max}) / {\cal D}(D_{min})$ explicitely in geometric resp. topologic terms of the base manifolds of the metric horns. We derive index formulas for the Spin-Dirac and Gau{\ss}-Bonnet operator. For the Signature operator we present a partial result. The first version of this paper was completed August 1995 at the University of Augsburg.