On two notions of curvature on singular surfaces

TL;DR

This paper proves the equivalence between two different notions of curvature bounds on singular surfaces, linking measure-based inequalities with classical Alexandrov curvature bounds, and explores implications for RCD spaces.

Contribution

It establishes the equivalence between measure-based curvature inequalities and Alexandrov curvature bounds on singular surfaces, filling a key gap in the theory.

Findings

Measure inequalities imply Alexandrov curvature bounds

Equivalence of curvature notions on singular surfaces

Lower measure-based bounds imply RCD(κ,2) condition

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 $\omega\geq\kappa\mu$ (resp. $\omega\leq\kappa\mu$) where $\omega$ is the curvature measure and $\mu$ 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 $\omega\geq\kappa\mu$ implies $RCD(\kappa, 2)$.