In 1959 Toponogov proved that on a closed surface of curvature at least 1, any simple closed geodesic has length at most 2π, with equality iff the surface is the round two-sphere. To prove this, one cuts the surface apart along the geodesic, and proves such a result for the boundary length of the resulting convex disks.
One might hope to prove a stability version of Toponogov’s theorem–a large geodesic means the surface is close to the round sphere–but tricky counterexamples exist! Motivated by the above proof of Toponogov’s theorem, however, we can nonetheless prove a stability version of the corresponding disk result, provided one now assumes strict convexity of the boundary. The argument uses a blend of theory from PDEs, complex analysis, and several notions of manifold convergence.