The structure of minimal surfaces in CAT(0) spaces

We prove that a minimal disc in a CAT(0) space is a local embedding away from a finite set of “branch points”. On the way we establish several basic properties of minimal surfaces: monotonicity of area densities, density bounds, limit theorems and the existence of tangent maps. As an application, we prove Fáry–Milnor’s theorem in the CAT(0) setting.