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Mathematics, Spectral Theory, math.SP,
Abstract:
The goal of this paper is to prove that on surfaces with asymptotically cusp
ends the relative determinant of pairs of Laplace operators is well defined. We
consider a surface with cusps (M,g) and a metric h on the surface that is a
conformal transformation of the initial metric g. We prove the existence of the
relative determinant of the pair $(\Delta_{h},\Delta_{g})$ under suitable
conditions on the conformal factor. The core of the paper is the proof of the
existence of an asymptotic expansion of the relative heat trace for small
times. We find the decay of the conformal factor at infinity for which this
asymptotic expansion exists and the relative determinant is defined. Following
the paper by B. Osgood, R. Phillips and P. Sarnak about extremal of
determinants on compact surfaces, we prove Polyakov's formula for the relative
determinant and discuss the extremal problem inside a conformal class. We
discuss necessary conditions for the existence of a maximizer.
