Compression theorems for surfaces and their applications

Abstract

Let $M$ be a complete glued surface whose sectional curvature is greater than or equal to $k$ and $\mathrm{▵}pqr$ a geodesic triangle domain with vertices $p,q,r$ in $M$. We prove a compression theorem that there exists a distance nonincreasing map from $\mathrm{▵}pqr$ onto the comparison triangle domain $\widetilde{\mathrm{▵}}pqr$ in the two-dimensional space form with sectional curvature $k$. Using the theorem, we also have some compression theorems and an application to a circular billiard ball problem on a surface.