Heat coefficients of surfaces with curved conic singularities

Let $(M,g)$ be a two-dimensional Riemannian manifold of finite diameter with a conical singularity. Under the assumption that the metric near the cone point $C$ is rotationally invariant, but not necessarily flat, we give an explicit formula for the coefficient $b_{1\mathrm{/}2}(C)$ in the heat trace expansion $\mathrm{tr}(\exp (-t{\mathrm{\Delta }}_g)){\sim }_{t\searrow 0}(4\pi t{)}^{-1}{\sum }_{j=0}^{\mathrm{\infty }}{a}_j(M)t^j+{\sum }_{j=0}^{\mathrm{\infty }}{b}_{j\mathrm{/}2}(C)t^{j\mathrm{/}2}+{\sum }_{j=0}^{\mathrm{\infty }}{c}_{j\mathrm{/}2}(C)t^{j\mathrm{/}2}\log t$. In the case that the Gaussian curvature $K$ of $(M,g)$ satisfies $\mathrm{\mid }K(p)\mathrm{\mid }\to \mathrm{\infty }$ as $p\to C$, we show that $b_{1\mathrm{/}2}(C)$ varies irrationally under constant rescalings of the distance circles near the cone point. This is a sharp contrast to the behavior of $b_0(C)$ and of those coefficients $b_j(C)$ which appear in certain known formulas in the case of orbifold cone points or corners of geodesic polygons.