The Zeta Functional Determinants on Manifolds with Boundary

✍ Scribed by Sun-Yung A. Chang; Jie Qing

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 420 KB
• Volume: 147
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1996.3059

✦ Synopsis

In this paper we derive some geometric formulas for the quotient of the zeta functional determinants for certain elliptic boundary value problems on Riemannian 3 and 4-manifolds with smooth boundary.

1997 Academic Press

1. Introduction

Let (M, g) denote a smooth compact Riemannian manifold without boundary. Via Weyl's invariant theory we know that all local scalar invariants of Riemannian geometry are linear combinations of monomials of covariant derivatives of Riemannian curvatures, which are constructed by pairing their indices and contracting to a scalar. A typical example, for instance, is the scalar curvature {. One can define the order of a local invariant as its homogeneity under a constant rescaling of the metric. Since the conformal class of metrics of a given metric g on M consists of all metrics e 2| g, where | is any smooth function on M, any integration of a local invariant becomes a functional over the function space C (M). One interesting class of such functionals over a given conformal class [ g] is given by the conformal primitives of local invariants in the following sense: article no.