Cisco Webex, Online seminar
(線上演講 Cisco Webex)
Allen--Cahn Equation and the Existence of Prescribed-mean-curvature Hypersurfaces
Neshan Wickramasekera (University of Cambridge)
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.
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Meeting number (access code): 184 238 9950