TITLE: Compact Eucledean hypersurfaces with "almost" constant H_k curvature
QUESTION [6 upvotes]: Let $M$ be an Eucledean $n$-dimensional compact hypersurface with constant $H_k$ curvature, where $k=1,...n$. A theorem by A.Ros tell us that so $M$ is an Eucledean sphere. Does anybody know if there are generalizations of such theorem?
In particular, I need something for Eucledean compact convex hypersurfaces that have $H_k$ curvature bounded and "almost" constant, in the sense that the measure of regions where $H_k$ is not constant is very small.
More precisely: let {$M_\epsilon$} be a family of convex compact Eucledean $n$-dimensional hypersurfaces depending by a parameter $\epsilon>0$. $M_\epsilon$ is $H_k$-constant except for a region whose measure is smaller than $\epsilon$. Furthermore we know that $H_k$ is bounded on $M_\epsilon$ uniformely respect to $\epsilon$.
Does exist some result that allows to say that when $\epsilon$ approaches to zero, then $M_\epsilon$ is "close" to a sphere?
A reference for A.Ros theorem is"Compact hypersurfaces with constant higher order mean curvatures." Rev. Mat. Iberoamericana 3 (1987), no. 3-4, 447–453.

REPLY [6 votes]: One can construct surfaces with mean curvature uniformly close to $1$, but look like collections of almost-touching spheres connected by perturbed catenoidal necks. The work of Ciraolo-Maggi (here) shows a certain stability result for these configurations. See the references therein for the constructions of these surfaces, and a discussion of other results (e.g. Hausdorff closeness to a single sphere with more hypotheses, like convexity or smallness of oscillation of mean curvature relative to the largest principal curvature).