Mathematics – Differential Geometry
Scientific paper
2010-10-19
Mathematics
Differential Geometry
Final version. To appear in International Journal of Mathematics. 20 pages
Scientific paper
Let P be a selfadjoint elliptic operator of order m>0 acting on the sections of a Hermitian vector bundle over a compact Riemannian manifold of dimension n. General arguments show that its zeta and eta functions may have poles only at points of the form s=k/m, where k ranges over all non-zero integers less than or equal to n. In this paper, we construct elementary and explicit examples of perturbations of P which make the zeta and eta functions be singular at all the points at which they are allowed to have singularities. We proceed within three classes of operators: Dirac-type operators, selfadjoint first-order differential operators, and selfadjoint elliptic pseudodifferential operators. As a result, we obtain genericity results for the singularities of the zeta and eta functions in those settings. In particular, in the setting of Dirac-type operators we obtain a new proof of a well known result of Branson-Gilkey, which was obtained by means of Riemannian invariant theory. As it turns out, the results of this paper contradict Theorem 6.3 of the third author's paper [Po1]. Corrections to that statement are given in Appendix B.
Loya Paul
Moroianu Sergiu
Ponge Raphael
No associations
LandOfFree
On the Singularities of the Zeta and Eta functions of an Elliptic Operator does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On the Singularities of the Zeta and Eta functions of an Elliptic Operator, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the Singularities of the Zeta and Eta functions of an Elliptic Operator will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-716716