  We consider a range of geometric stability problems for hypersurfaces of
spaceforms. One of the key results is an estimate relating the Hausdorff
distance to a geodesic sphere of an embedded hypersurface to the $L^{2}$-norm
of the traceless Hessian operator of a defining function for the open set
bounded by the hypersurface. As application, this result allows for a unified
treatment of many old and new stability problems arising in geometry and
analysis. Those problems ask for spherical closeness of a hypersurface, given a
geometric constraint. Examples include stability in Alexandroff's soap bubble
theorem in space forms, Serrin's overdetermined problem, a Steklov problem
involving the bi-Laplace operator and non-convex Alexandroff-Fenchel
inequalities.
