First steps towards a proof for the renormalizability of scalar \phi^{4} theories on Riemannian manifolds with boundaries

The long term goal of my project is to prove the renormalizability of non minimally coupled scalar phi⁴ theories on curved Riemannian Manifolds with boundaries. The method used in establishing perturbative renormalization of such models is based on the renormalization group flow equations. It is mainly inspired by the renormalizability proof of the scalar phi⁴ theory on a Riemannian half space by Kopper and Borji. Their proof requires a detailed knowledge of the propagators, or equivalently the heat kernels of the underlying theory. Hence, a first significant step of my work is to obtain an accurate understanding of the heat kernels in generically curved and compact Riemannian manifolds with boundaries, as well as their properties. In this talk I will give a brief overview on how one can obtain a local expansion of the heat kernel in terms of the curvature, discuss several of their properties and, most importantly, what one would need in order to generalize the proof of Kopper and Borji to the case of curved manifolds with boundary.