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 $¥triangle$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 $¥triangle$pqr onto the comparison triangle domain $¥widetilde{¥triangle}$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.