do Carmo conjecture on stable constant-mean-curvature hypersurfaces

Prove that every constant-mean-curvature hypersurface in \(\mathbb{R}^{n+1}\) that passes the second derivative test for area among all hypersurfaces enclosing the same relative volume is a round sphere for every \(n\geq 2\).

Background

The paper compares its rigidity theorem for weakly stable Serrin domains with a conjecture attributed to Manfredo do Carmo. That conjecture asserts an analogous rigidity phenomenon for constant-mean-curvature hypersurfaces under volume-preserving stability. It is mentioned as a related unresolved statement and is not proved by the results of the paper.

References

We note that, in a similar context, M.~do Carmo has conjectured that constant mean curvature hypersurfaces in $\mathbb{R}{n+1}$ that pass the second derivative test for area among all hypersurfaces enclosing the same relative volume are round spheres for all $n\geq 2$.

— Weakly stable solutions of Serrin's problem  (2608.19986 - Eichmair et al., 20 Aug 2026) in Introduction