The structure of minimal surfaces in CAT(0) spaces

TL;DR

This paper investigates the properties of minimal surfaces in CAT(0) spaces, establishing foundational results and applying them to prove a classical knot theory theorem within this geometric context.

Contribution

It introduces new structural insights into minimal surfaces in CAT(0) spaces, including local embedding properties and tangent map existence, extending classical results to non-Euclidean settings.

Findings

Minimal discs are locally embeddings except at finitely many branch points.

Established monotonicity and density bounds for minimal surfaces.

Proved Fary-Milnor's theorem in the CAT(0) setting.

Abstract

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 Fary-Milnor's theorem in the CAT(0) setting.