Moderate smoothness of most Alexandrov surfaces

TL;DR

Most Alexandrov surfaces with curvature bounded below are smooth at most points, lacking conical singularities, and exhibit a specific curvature behavior where lower and upper Gaussian curvatures are respectively equal to  and infinity.

Contribution

This paper demonstrates that, in a Baire category sense, most Alexandrov surfaces are free of conical points and characterizes their Gaussian curvature properties.

Findings

Most Alexandrov surfaces have no conical points.

At most points, lower curvature equals  and upper curvature is infinite.

The result applies to surfaces with curvature bounded below by .

Abstract

We show that, in the sense of Baire category, most Alexandrov surfaces with curvature bounded below by $\kappa$ have no conical points. We use this result to prove that at most points of such surfaces, the lower and the upper Gaussian curvatures are equal to $\kappa$ and $\infty$ respectively.