Katrin Wendland's Diploma Thesis — Summary
Analytic Torsion and Critical Metrics
Ray-Singer Analytic Torsion, Quillenmetric and Regularized Determinants
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
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
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
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