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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.

Stability for Serrin’s Problem and Alexandroff’s Theorem in Warped Product Manifolds