Zeta Functional Determinants on manifolds with boundary

Abstract

This article is an announcement of our recent work on the zeta functional determinants on manifolds with boundary.We first derive some geometric formulas for the quotient of the zeta functional determinants for certain elliptic boundary value problems on Riemannian 3 & 4-manifolds with smooth boundary.We then apply the formulas to establish W 2,2compactness of isospectral set within a subclass of conformal metrics, and to prove some existence and uniqueness properties of extremal metrics for the zeta functional determinants.Some key elements in our proof include the discovery of some boundary operator conformal covariant of degree 3 and establishment of some sharp Sobolev trace inequalities of Lebedev-Milin type. Part 1. Alvarez-Polyakov formula PreliminariesLet M be a compact Riemannian manifold with smooth boundary ∂M .We shall consider boundary value problems (A, B) satisfying the three basic assumptions below.For simplicity we shall only consider operators acting on the space of functions on M , although all the statements in this article could be formulated for operators acting on the space of sections of a tensorspinor bundle over M .Analytic Assumptions.Let A be a differential operator of order 2l.Suppose that A is formally self-adjoint and has positive definite leading symbol.Let B be an operator for the Cauchy data for A on ∂M such that the pair (A, B) is elliptic.

Locations

• Mathematical Research Letters - View - PDF