Flats and the flat torus theorem in systolic spaces

Abstract

We prove the Systolic Flat Torus Theorem, which completes the list of basic properties that are simultaneously true for systolic geometry and $CAT\left(0\right)$ geometry.

We develop the theory of minimal surfaces in systolic complexes, which is a powerful tool in studying systolic complexes. We prove that flat minimal surfaces in a systolic complex are almost isometrically embedded and introduce a local condition for flat surfaces which implies minimality. We also prove that minimal surfaces are stable under small deformations of their boundaries.