Analysis, Spectra and Number Theory, a Conference in Honor of Peter Sarnak’s 61st Birthday

In a series of joint papers in 1988-89, Osgood-Phillips-Sarnak identified the extremal metrics of the zeta functional determinant of the Laplacian operator on compact surfaces and applied the result to establish the compactness of isospectral sufaces. In this talk, I will report on extending these ideas to study some corresponding functional determinant in three and four dimensional manfolds under conformal change of metrics, tools from conformal geometry to study the functional, the extremal metrics on model cases and the corresponding (sharp) Sobolev trace inequalites.