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.
Yau.Problem section.InSeminar on differential geometry, volume 102 ofAnnals of Mathematics Studies, Princeton University Press
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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.