A heat trace anomaly on polygons
Rafe Mazzeo, Julie Rowlett · Mathematical Proceedings of the Cambridge Philosophical Society · 2015
Abstract Let Ω0be a polygon in $\mathbb{R}$ 2, or more generally a compact surface with piecewise smooth boundary and corners. Suppose that Ωεis a family of surfaces with ${\mathcal C}$ ∞boundary which converges to Ω0smoothly away from the corners, and in a precise way at the vertices to be described in the paper. Fedosov [6], Kac [8] and McKean–Singer [13] recognised that certain heat trace coefficients, in particular the coefficient oft0, are not continuous as ε ↘ 0. We describe this anomaly using renormalized heat invariants of an auxiliary smooth domainZwhich models the corner formation. The result applies to both Dirichlet and Neumann boundary conditions. We also include a discussion of what one might expect in higher dimensions.