Differentiable stability and sphere theorems for manifolds and Einstein   manifolds with positive scalar curvature
  Leon Green obtained remarkable rigidity results for manifolds of positive scalar curvature with large conjugate radius and/or injectivity radius. Using $C^{k,\alpha}$ convergence techniques, we prove several differentiable stability and sphere theorem versions of these results and apply those also to the study of Einstein manifolds. 
Wilderich Tuschmann and Michael Wiemeler
math.DG