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The goal of this paper is to prove that on surfaces with asymptotically cuspends the relative determinant of pairs of Laplace operators is well defined. Weconsider a surface with cusps (M,g) and a metric h on the surface that is aconformal transformation of the initial metric g. We prove the existence of therelative determinant of the pair $(\Delta_{h},\Delta_{g})$ under suitableconditions on the conformal factor. The core of the paper is the proof of theexistence of an asymptotic expansion of the relative heat trace for smalltimes. We find the decay of the conformal factor at infinity for which thisasymptotic expansion exists and the relative determinant is defined. Followingthe paper by B. Osgood, R. Phillips and P. Sarnak about extremal ofdeterminants on compact surfaces, we prove Polyakov's formula for the relativedeterminant and discuss the extremal problem inside a conformal class. Wediscuss necessary conditions for the existence of a maximizer.

Asymptotics of relative heat traces and determinants on open surfaces of finite area