Continuity of relative hyperbolic spectral theory through metric degeneration
The authors discuss the Minakshisundaram-Pleijel pair of punctured Riemann surfaces in the following sense. Let \(M_1\) and \(M_2\) be compact Riemann surfaces. Let \(S_1\), \(S_2\) be finite subsets of \(M_1\) and \(M_2\), respectively. Let the metrics \(\mu_1\), \(\mu_2\) realise the conformal structures on \(M_1-S_1\) and \(M_2-S_2\). It is assumed that \(S_1\) and \(S_2\) have the same cardinality and that the metrics ``agree'' on certain neighbourhoods of the points \(S_1\) and \(S_2\). It is then possible to define a zeta function of the ``difference'' of these two objects. The authors use the theory of the heat kernel to investigate how the zeta function varies as the metric varies, and, especially, when the variation either introduces or erases cusps, i.e. points of \(S_1\) and \(S_2\) where the metric becomes singular in the same way as a hyperbolic metric does. They then apply these results to the study of the ``logarithm of a regularised determinant of the Laplacian'' and the closely related structure constant. The authors show that under certain circumstances the Polyakov integrals cancel and that consequently these quantites are constant under the deformation and equal those of the limiting objects.
scientific article; zbMATH DE number 922153
scientific article