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 $b_{1/2}(C)$ in the heat trace expansion $\operatorname{tr}(\operatorname{exp}(-t\Delta_g))\sim_{t\searrow0} (4\pi t)^{-1}\sum_{j=0}^\infty a_j(M) t^j+\sum_{j=0}^\infty b_{j/2}(C)t^{j/2}+\sum_{j=0}^\infty c_{j/2}(C) t^{j/2} \log t$. In the case that the Gaussian curvature $K$ of $(M,g)$ satisfies $|K(p)|\to\infty$ as $p\to C$, we show that $b_{1/2}(C)$ 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.