##### Department of Mathematics,

University of California San Diego

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### MATH 258 - Differential Geometry Seminar

## Neshan Wickramasekera

#### University of Cambridge

## Allen--Cahn equation and the existence of prescribed-mean-curvature hypersurfaces

##### 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 $N$ of dimension $n \geq 3$ and any non-negative (or non-positive) Lipschitz function $g$ on $N$, there is a boundaryless $C^{2}$ hypersurface $M \subset N$ whose scalar mean curvature is prescribed by $g.$ More precisely, the hypersurface $M$ is the image of a quasi-embedding $\iota$ (of class $C^{2}$) admitting a global unit normal $\nu$ such that the mean curvature of $\iota$ at every point $x$ is $g(\iota(x))\nu(x)$. Here a 'quasi-embedding' is an immersion such that any point of its image near which the image is not embedded has an ambient neighborhood in which the image is the union of two $C^{2}$ embedded disks with each disk lying on one side of the other (so that any self-intersection is tangential). If $n \geq 7$, the singular set $\overline{M} \setminus M$ may be non-empty, but has Hausdorff dimension no greater than $n-7$. 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. It utilises the elliptic and parabolic Allen-Cahn equations on $N$, and brings to bear on the question 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 alternative to 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, a ''black-box'' tool of independent interest (also joint work with Bellettini). This theory provides multi-sheeted $C^{1, \alpha}$ regularity for mean-curvature-controlled codimension 1 integral varifolds $V$ near points where one tangent cone is a hyperplane of multiplicity $q \geq 2;$ this regularity holds whenever: (i) $V$ has no classical-singularities, i.e. no portion of $V$ is the union of three or more embedded hypersurfaces-with-boundary coming smoothly together along their common boundary, and (ii) the region where the mass density of $V$ is $< q$ is 'well-behaved' in a certain topological sense. A very important feature of this theory, crucial for its application to the Allen--Cahn method, is that $V$ is not assumed to be a critical point of a functional.

### April 28, 2022

### 11:00 AM

Zoom ID: 924 6512 4982

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