%0 Journal Article
%@ 1050-6926
%A Bellettini, Costante
%A Marshall-Stevens, Kobe
%D 2025
%F discovery:10205629
%I Springer Verlag
%J Journal of Geometric Analysis
%T On isolated singularities and generic regularity of min-max CMC hypersurfaces
%U https://discovery.ucl.ac.uk/id/eprint/10205629/
%X In compact Riemannian manifolds of dimension 3 or higher with positive Ricci  curvature, we prove that every constant mean curvature hypersurface produced by  the Allen-Cahn min-max procedure of Bellettini-Wickramasekera (with constant  prescribing function) is a local minimiser of the natural area-type functional  around each isolated singularity. In particular, every tangent cone at each  isolated singularity of the resulting hypersurface is area-minimising. As a  consequence, for any real $\lambda$ we show, through a surgery procedure, that  for a generic 8-dimensional compact Riemannian manifold with positive Ricci  curvature there exists a closed embedded smooth hypersurface of constant mean  curvature $\lambda$; the minimal case ($\lambda$ = 0) of this result was  obtained in work by Chodosh-Liokumovich-Spolaor.
%Z This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.