Asymptotic expansion of relative heat traces on surfaces with asymptotic cusps

Asymptotic expansions of heat traces appear naturally in the study of spectral invariants and their variations. In this talk I consider a surface with cusps and a conformal variation of the metric. The conformal factor does not necessarily have compact support but should decay at infinity. I consider the Laplace operators associated to these metrics and the corresponding heat operators. If the conformal factor decays at infinity, the difference of the heat operators is trace class. I will give explicit conditions on the decay of the conformal factor that, in addition, allow the existence of an asymptotic expansion of this trace (up to certain order) for small values of t.