Gauss-Bonnet on a finite, non-compact surface.

Question

Let $D$ be a two dimensional (simply connected) compact surface with boundary and a consider a point $p\in D$. Since $D \backslash\{p\}$ is no longer compact, the Gauss-Bonnet identity does not immediately apply, then is there any Gauss-Bonnet-like global constraint on the curvature of $D \backslash \{p\}$ (or in general, about $D \backslash\{\text{measure zero set}\}$) ?

Answer 1

A surface with a cone point may be viewed as having infinite curvature at the vertex, in such a way that the integral of the curvature at the vertex is equal to $2\pi$ minus the incident angle. The Gauss-Bonnet theorem holds for such surfaces.

If $C > 0$ is arbitrary, there exists a smooth, flat metric on the punctured unit disk for which the total geodesic curvature of the boundary is $C$. (In fact, these can be embedded isometrically in flat Euclidean $3$-space as cones with arbitrary positive angle at the vertex, except the vertex is removed.)

Because we can concentrate curvature at a vertex (maintaining the Gauss-Bonnet theorem) then remove the vertex (subtracting $2\pi - C$ from one side), there is no Gauss-Bonnet theorem for punctured surfaces.