%O This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
%D 2025
%I Springer Verlag
%L discovery10205629
%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.
%J Journal of Geometric Analysis
%A Costante Bellettini
%A Kobe Marshall-Stevens
%T On isolated singularities and generic regularity of min-max CMC hypersurfaces