On index formulas for manifolds with metric horns. (English) Zbl 0906.58049

Lesch, Matthias; Peyerimhoff, Norbert

Commun. Partial Differ. Equations 23, No. 3-4, 649-684 (1998).

In this paper it is discussed the index problem for geometric differential operators (spin-Dirac operator, Gauß-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 \(D_{\max}\). The authors describe the quotient \({\mathcal D}(D_{\max})/{\mathcal D}(D_{\min})\) explicitly in geometric resp. topological terms of the base manifolds of the metric horns. Moreover, they derive index formulas for the spin-Dirac operator and for the Gauß-Bonnet operator. Some partial results are presented for the signature operator.

Reviewer: Lubomira Softova (Sofia)

MSC:

58J20	Index theory and related fixed-point theorems on manifolds
58J05	Elliptic equations on manifolds, general theory

Keywords:

index; spin-Dirac operator; Gauß-Bonnet operator; signature operator; manifolds with metric horns