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.
Gromov.In Dimension, nonlinear spectra and width.InGeometric aspects of functional analysis (1986/87),volume 1317 ofLecture Notes in Math.,1317, 132–184, Springer, Berlin
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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.