In my diploma thesis, I investigated regularized determinants of elliptic differential operators. They can be used to construct interesting functionals on the moduli space of Kähler metrics of a complex Kähler manifold, for example by the seminal results of Osgood, Phillips and Sarnak. The celebrated anomaly formula of Bismut, Gillet and Soulé for the curvature of the determinant line bundle has been a main ingredient of my studies concerning the dependence of Ray-Singer analytic torsion and Quillen metric on the metric of the underlying manifold. With my diploma supervisor Prof. Werner Müller, I investigated critical points of these functionals:
E.g. for polarized algebraic manifolds (X,L), we use a twisted version of the Quillen metric and prove that its critical points are the metrics of constant scalar curvature. In particular, for c1(X)=0 or c1(X)<0 this gives a new characterization of Kähler-Einstein metrics as absolute maxima of our functional. This is analogous to Donaldson's result that the Hermite-Yang-Mills metric of a holomorphic bundle with vanishing first Chern class can be characterized as a critical point of another twisted version of the Quillen metric. However, we use the evolution of the metric by the complex analog of Hamilton's Ricci flow equation for our proof. A generalization to metrically polarized families of compact Hodge manifolds along the lines of Fujiki and Schumacher is possible. Here, our functional is a potential of the generalized Weil-Petersson metric.
