preprint

Inserted: 17 aug 2026

Year: 2026

Abstract:

We prove several analytic results on (possibly noncompact) complete singular surfaces having locally bounded integral curvature (in short: BIC surfaces). Regarding these as metric measure spaces with the 2-dimensional Hausdorff measure, we show that these are infinitesimally Hilbertian, locally doubling and satisfy a local Poincaré inequality. In particular, this entails the existence of a jointly Hölder continuous heat kernel for the Cheeger Laplacian. Assuming that the negative part of the curvature measure of a BIC surface satisfies a Dynkin-type condition, we show that the surface is bi-Lipschitz equivalent to a BIC surface with a lower bounded curvature measure, entailing global variants of the aforementioned results.