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- > Prof. Dr. Katrin Wendland

My diploma thesis contains the following results:

- I generalized B. Osgood's, R. Phillips' and P. Sarnak's methods concerning the determinant of the Laplacian on compact Riemann surfaces to noncompact ("admissible") Riemann surfaces. In doing so I was able to generalize the definition of the determinant to metrics which belong to the conformal class of an admissible metric and are "near by" an admissible metric in every cusp. For Riemann surfaces with cusps I proved the existence of a unique metric of constant Gauß curvature in every conformal class of such metrics of given volume. Moreover, I showed that the metric of constant Gauß curvature is the unique critical point of the determinant of the Laplacian in any such class of metrics, namely a maximum.
- Using known formulae [BGSIII] I computed and solved the Euler-Lagrange equations for critical points of the determinant of the Laplacian on closed Riemann surfaces under variation of the metric within the Kähler class. Again the metrics of constant Gauß curvature are the critical points. Moreover I showed, that for Riemann surfaces with nonpositive Euler characteristics these critical points are strict local maxima, and that they agree with the critical points of the Quillenmetric for Riemann surfaces. Thus, I have found a second independent proof for some of the well known results of B. Osgood's, R. Phillips' and P. Sarnak's for closed Riemann surfaces.
- On compact manifolds which admit a metric of constant holomorphic curvature I showed that metrics of constant holomorphic curvature are critical with respect to the Quillenmetric under variation of the metric on the underlying manifold within its Kähler class. This result was achieved by using the so-called anomaly formula [BGSIII].
- I found an elementary proof for the variational formula of the analytic torsion near a Kähler-Einstein metric under variation within the Kähler class, which may be applied to closed K3-surfaces and complex two dimensional tori. For K3-surfaces this formula leads to a criterion for the curvature tensor in order to decide whether a Kähler-Einstein metric is critical with respect to the analytic torsion under variation within the Kähler class. For complex tori of dimension N=2 it leads to an elementary proof for the fact that flat metrics are critical points of the analytic torsion under variation of the metric within the Kähler class. For K3-surfaces and tori I found that the results for the variation of analytic torsion and Quillenmetric around a Kähler-Einstein metric must agree.
- Besides the above results on manifolds with constant holomorphic curvature I proved that for complex tori of dimension N≥2 the second variation of the Quillenmetric vanishes in the flat metric, and that flat metrics are critical with respect to the analytic torsion. These results again were achieved using known formulae. From the results stated above one can deduce that as generalization of the determinant of the Laplacian on Riemann surfaces the Quillenmetric is more adequate than the analytic torsion.
- Concerning the variation of the Quillenmetric on determinant line bundles corresponding to families of Dolbeaut-Dirac operators on forms with values in holomorphic vector bundles over closed algebraic manifolds I found line bundles such that a variation of metric on the manifold and on the bundle is possible simultaneously. I showed that after having chosen appropriate bundles the metrics of vanishing scalar curvature are the critical points under this variation. I can decide what type of critical point a Ricci flat Kähler-Einstein metric is.

Moreover, summaries of already known results which concern topics of my diploma thesis may be found within the thesis. The last chapter is devoted to connections with various topics in theoretical physics. In some cases I showed, that statements and methods given in the physical literature are identical with those given in mathematical literature, at least up to (physically) irrelevant terms.

A lot of questions had to remain unanswered, among them the following:

- What kind of information about the spectrum of the Laplacian corresponding to a "conformally admissible" metric is given by the "g-scaling determinant" of definition 5.1.11? Can it be interpreted as relative determinant of the Laplacian with respect to an appropriate operator?
- What kind of critical points are the metrics of constant holomorphic curvature and the flat metrics on complex tori?
- What kind of critical points are the metrics of vanishing scalar but nonvanishing Ricci curvature under the variation of the metric on holomorphic bundles over algebraic manifolds described above?
- Are Ricci flat Kähler-Einstein metrics on closed K3-surfaces critical with respect to the analytic torsion or the Quillenmetric under variation within their Kähler class?

The answer to the first question would exceed the scope of a diploma thesis. Neither the second nor the third question could be answered by mere calculations with local coordinates up to now. Nor did an ansatz in the spirit of K. Richardson's ideas (chapter 5.2) lead to any further insight. The answer to question number four seems not to be elementary, one possibility of further investigation was given at the end of chapter 6.9.

*for further information on the contents of this work see its table of contents*