Conformal Perturbation of Heat Kernels and Their Traces

On a compactn-dimensional Riemannian manifoldMwe consider self-adjoint operators of typew−1Δacting onL2(M, w dx), wheredxis the volume element ofMandw>0 smooth. Forn=2 this setup arises naturally from a conformal change of metric. We derive perturbation series for the heat kernels of these operators and their traces, in terms ofwand of objects in the given background metric. The method involves Gaussian estimates on space and time derivatives of the complex time heat kernel.