abstract: Conformal variations of renormalized determinants for covariant operators can be sometimes expressed by explicit formulas. It is natural then to find metrics which might extremize the determinants: for example in two dimensions this is a way to characterize constant curvature metrics. I will discuss the derivation of such formulas and some cases in which it is possible to find or characterize extremal metrics.
The course will consist of four lectures, during about one hour and a half each. The material covered in the lectures will be the following: 1. Review on the min-max formulas for eigenvalues of the Laplacian, Weyl's asymptotic formula, heat kernels and the Minakshisundaram-Pleijel expansions. 2. Functional determinant of the Laplacian, Ray-Singer-Polyakov formulas, characterization of extremal metrics by Osgood-Phillips-Sarnak, Onofri inequality. 3. Isospectral surfaces, higher order heat invariants and compactness results for isospectral metrics in high order norms. 4. Functional determinants in higher dimensions, compactness of isospectral metrics, determinants of conformally invariant operators, existence and uniqueness of extremal metrics.
The course will require some basic knowledge and familiarity with Riemannian geometry, Sobolev spaces and regularity theory for elliptic partial differential equations.