Speaker
Description
It is known that constant mean curvature (CMC) hypersurfaces, with or without boundary, in a smooth Riemannian manifold $(M, g)$ can develop singularities if $n\geq 7$. For a CMC hypersurface $S$ that minimises the prescribed mean curvature functional in a ball $B_r(0)$ with boundary $\partial S$ in $\partial B_r(0)$ and an isolated singularity at the origin we construct in enter link description here a sequence $S_j$ of smooth CMC hypersurfaces such that $S_j \to S$ in the Hausdorff distance sense. We employ techniques from both Geometric measure theory and Elliptic PDE theory to set up a perturbation problem and first construct a sequence of possibly singular CMC hypersurfaces that converge to the initial hypersurface $S$. In order to prove that these hypersurfaces are actually smooth we develop a singular maximum principle. This then allow us to blow-up along the possible singularities of the constructed sequence and come to a contradiction using the well-known Hardt-Simon foliation for area minimising tangent cones with isolated singularities.
