Stability of mean convex cones under mean curvature flow
  We consider graphical solutions to mean curvature flow and obtain a stability result for homothetically expanding solutions coming out of cones of positive mean curvature: If another solution is initially close to the cone at infinity, then the difference to the homothetically expanding solution becomes small for large times. The proof involves the construction of appropriate barriers. 
Julie Clutterbuck and Oliver C. Schn\"urer