On two notions of curvature on singular surfaces
Maxime Marot
TL;DR
The paper establishes an equivalence between curvature bounds given by the curvature measure $ω$ against the area measure $μ$ on BIC surfaces and classical Alexandrov bounds, showing $ω ≥ κ μ$ yields $CBB(κ)$ and $ω ≤ κ μ$ yields locally $CAT(κ)$ (with κ ≤ 0). The core method translates curvature-area distortions into precise hinge-angle comparisons, leveraging area distortion inequalities and Gauss–Bonnet arguments to bridge the two frameworks. A key corollary, via Petrunin’s results, is that $(S, μ)$ satisfies the $RCD(\kappa,2)$ condition when $ω ≥ κ μ$ (with mild cusp assumptions). The work fills a gap by proving the forward implications for arbitrary κ and clarifies the sharpness of local curvature bounds on singular surfaces, with implications for the structure of metric-measure spaces in this setting.
Abstract
In this paper, we investigate the equivalence of two distinct notions of curvature bounds on singular surfaces. The first notion involves inequalities of the form $ω\geqκμ$ (resp. $ω\leqκμ$) where $ω$ is the curvature measure and $μ$ the Hausdorff measure. The second notion is the classical Alexandrov curvature bound CBB (resp. CAT). We demonstrate that these two definitions are, in fact, equivalent. Specifically, we fill an important gap in the theory by showing that the inequalities imply the corresponding Alexandrov CBB (resp. CAT) bound. One striking application of our result is that, in combination with a result of Petrunin, the lower bound $ω\geqκμ$ implies $RCD(κ, 2)$.