Abstract:

The lecture will discuss recent joint work with Costante Bellettini at UCL. A main outcome of the work is a proof that for any closed Riemannian manifold of dimension and any non-negative (or non-positive) Lipschitz function on , there is a boundaryless hypersurface whose scalar mean curvature is prescribed by The hypersurface is the image of a quasi-embedding $\iota$ (of class ) admitting a global unit normal such that the mean curvature of at every point is . Here a `quasi-embedding' is an immersion such that any point of its image where the image is not embedded has an ambient neighborhood in which the image is the union of two embedded disks with each disk lying on one side of the other (and hence any self-intersection tangential). If , the singular set may be non-empty, but has Hausdorff dimension no greater than . An important special case is the existence of a CMC hypersurface for any prescribed value of mean curvature. The method of proof is PDE theoretic and utilises the elliptic and parabolic Allen--Cahn equations on . It brings to bear on the question certain elementary, and yet very effective, variational and gradient flow principles in semi-linear elliptic and parabolic PDE theory---principles that serve as a conceptually and technically simpler replacement for the Geometric Measure Theory machinery pioneered by Almgren and Pitts to prove existence of a minimal hypersurface. For regularity conclusions the method relies on a new varifold regularity theory of independent interest (also joint work with Bellettini). This theory provides multi-sheeted regularity for mean-curvature-controlled codimension 1 integral varifolds near points where one tangent cone is a hyperplane of multiplicity this regularity holds whenever (i) no portion of is the union of three or more hypersurfaces-with-boundary coming smoothly together along their common boundary, and (ii) the region where the mass density of is is `well-behaved' in a certain topological sense. A very important feature of this theory is that is not assumed to be a critical point of any functional.