On manifolds with finitely many c-CMC hypersurfaces, a nearby metric exists that preserves the original ones while adding strictly more c-CMC hypersurfaces, with an explicit bound on the metric change in L^{(n+1)/2} norm.
Simon.Lectures on geometric measure theory.Proceedings of the Centre for Mathematical Analysis, Australian National University, 3
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Existence of multiple constant mean curvature hypersurfaces for varying Riemannian metrics
On manifolds with finitely many c-CMC hypersurfaces, a nearby metric exists that preserves the original ones while adding strictly more c-CMC hypersurfaces, with an explicit bound on the metric change in L^{(n+1)/2} norm.