  We prove quantitative versions for several results from geometric partial
differential equations. Firstly, we obtain a double stability theorem for
Serrin's overdetermined problem in spaceforms. Secondly, we prove stability
theorems for Brendle's Heintze-Karcher inequality respectively constant mean
curvature classification in a class of warped product spaces. The key tool is
the first author's recent development of stability for level sets of a function
under smallness of the traceless Hessian thereof.
