Quantitative stability for hypersurfaces with almost constant mean   curvature in the hyperbolic space
  We provide sharp stability estimates for the Alexandrov Soap Bubble Theorem in the hyperbolic space. The closeness to a single sphere is quantified in terms of the dimension, the measure of the hypersurface and the radius of the touching ball condition. As consequence we obtain a new pinching result for hypersurfaces in the hyperbolic space.   Our approach is based on the method of moving planes. In this context we carefully review the method and we provide the first quantitative study in the hyperbolic space. 
Giulio Ciraolo and Luigi Vezzoni
math.DG math.AP