By the solution of Plateau's problem, every simple closed curve in Euclidean space bounds an immersed minimal surface of disk-type.
If the total curvature of the curve is at most $4\pi$ then this minimal disk is embedded. In the talk I will discuss a generalization
of this embeddedness result for minimal disks in Riemannian manifolds of sectional curvature bounded above. The proof relies on
techniques from metric geometry, developed for the study of minimal surfaces in non-smooth ambient spaces, and applies more generally
for CAT(K) spaces.
This is joint work with Stephan Stadler.