Heat coefficients of surfaces with curved conical 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 b1/2(C)in the heat trace expansion tr(exp(-tΔg))∼t↘0(4πt)-1∑j=0∞aj(M)tj+∑j=0∞bj/2(C)tj/2+∑j=0∞cj/2(C)tj/2logt. In the case that the Gaussian curvature K of ( M , g ) satisfies |K(p)|→∞as p→C, we show that b1/2(C)varies irrationally under constant rescalings of the distance circles near the cone point. This is a sharp contrast to the behavior of b0(C)and of those coefficients bj(C)which appear in certain known formulas in the case of orbifold cone points or corners of geodesic polygons.