Zeta functional determinants on manifolds with boundary

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^2,2-compactness 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.