Phenomena in Spectral Theory

The extremal problem for the functional determinant of a natural linear elliptic operator a on Riemannian manifold is studied. Viewing the determinant as a function of the Riemannian metric, we encounter non-linear geometric analytic phenomena: sharp inequalities comparing nonlinear functionals of the metric and its derivatives. The derivation and use of such inequalities in new situations, especially essentially tensor-valued inequalities, leads back to linear theory and the classiication of conformally covariant differential operators. 0. Introduction A central object of study in Geometric Analysis is the space G(M) of Rie-mannian metrics on a smooth compact manifold M. The most revealing data in this study are the spectra of diierential operators A g which are functorially, or naturally , associated to the metric g; for example, the Laplacian. The diieomorphism group Diieo(M) is the gauge transformation group in this setting; the spectrum of a natural A g will be unaaected by diieomorphisms and thus gauge invariant. The multiplicative group C 1 + (M) of smooth positive real functions e ! on M also acts on G(M) in a natural way, by conformal change g 7 ! e 2! g. The space G(M)= Diieo(M) can thus be broken down into several parts (and hopefully reassembled in the long run): the quotient G(M)=(Diieo(M) n C 1 + (M)), the conformal classes C 1 + (M)g, and the intersection of the two group actions: conformal changes that are actually implemented by diieomorphisms, the conformal transformation group C(M; g). The functional determinant det A g is a spectral invariant that is apparently quite revealing of the geometry of g. Originally of interest in quantum eld theory, where it provides a regularization of the functional integral, the functional determinant has recently become the object of intense study in connection with String Theory in Physics, and the isospectral problem in Mathematics. These two pursuits illustrate complementary approaches to an understanding of the space of metrics: (1) one can do an extremal problem, to try to get a representative of a conformal class which is somehow \uniform"; and (2) one can use spectral invariants to bound the metric, showing that there is a unique or nearly unique metric (modulo gauge)