The relative determinant and the function xi (Q2709146)

This paper is a short summary of the author's paper ``Déterminant relatif et la fonction xi'' to appear at the American Journal of Mathematics. NEWLINENEWLINENEWLINEIf a Riemannian manifold \((M,g)\) is complete, but not compact, it is difficult to define the determinant of the Laplacian. The author defines a relative determinant (determinant on \(M\) relative to a complement of an open bounded set \({\mathcal O}\subset M\)).NEWLINENEWLINENEWLINEThe first result relates this relative determinant to the determinant of the Dirichlet to Neumann operator on \(C^\infty(\partial{\mathcal O})\).NEWLINENEWLINENEWLINEThe second result relates the relative determinant to the function of spectral shift. NEWLINENEWLINENEWLINEThe behaviour of this spectral shift function to infinity is given.NEWLINENEWLINEFor the entire collection see [Zbl 0955.00015].