We construct the heat kernel on the setting of manifolds with fibred boundary metrics. In physics, these metrics arise as gravitational instantons or complete hyperkahler four-manifolds. For such manifolds, we analyze the asymptotics of heat kernels in finite and long time. The construction approach is the $\phi$-calculus of Melrose and Mazzeo combined with the resolvent construction of Guillarmou and Hassell. We present the analytic torsion of such manifolds as an application of these constructions and impose the natural question of how to relate this to the topological invariant, i.e. Reidemeister Torsion in this setting.

Keywords: Heat kernel, Resolvent at low energy, fibred boundary manifolds, Analytic torsion

Reference: http://oops.uni-oldenburg.de/4772/1/talspe20.pdf