On index formulas for manifolds with metric horns

TL;DR

This paper investigates the index problem for key geometric differential operators on manifolds with metric horns, providing explicit descriptions of operator extensions and deriving index formulas, with partial results for the Signature operator.

Contribution

It introduces explicit descriptions of extremal operator extensions and derives index formulas for Spin-Dirac and Gau{ extbackslash}ss-Bonnet operators on singular manifolds with metric horns.

Findings

Explicit description of the quotient of operator domains.

Index formulas for Spin-Dirac and Gau{ extbackslash}ss-Bonnet operators.

Partial index results for the Signature operator.

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.