16:15 Uhr | Vlad Moraru (Warwick) | On Area Comparison and Rigidity Involving the Scalar Curvature
I shall describe an area comparison theorem for certain totally geodesic
surfaces in 3-manifolds with lower bounds on the scalar curvature. This result
is an optimal analogue of the Heintze-Karcher-Maeda area comparison theorem for
minimal hypersurfaces in manifolds of non-negative Ricci curvature. I shall then
show how this area comparison theorem provides a unified proof of three splitting
and rigidity theorems for 3-manifolds with lower bounds on the scalar curvature that
were first proved, independently, by Cai-Galloway, Bray-Brendle-Neves and Nunes. This is
joint work with Mario Micallef. Finally, I shall address some natural higher dimensional
generalisations of these area comparison and rigidity results.bstract
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17:45 Uhr | Vincent Feuvrier (Toulouse) | Approximating solutions to the Plateau problem using uniform polyhedric granulometry
We consider generic measure-minimization problems with weak
initial assumptions on the regularity of competitors (we do not suppose orientability nor even
rectifiability). A subset of $\mathbf R^n$ is said to be minimal if its $d$-dimensional Hausdorff
measure cannot be decreased by a deformation taken in a suitable homotopy class. The classic Plateau
problem can be rewritten in these terms by finding a minimal set under deformations that only move a
relatively compact subset of points of a given domain: in that case the boundary of the domain acts
as a topological constraint. There are relatively few existence results under this setup, compared to
classical approaches based upon differential geometry.
We provide a method to convert any minimizing sequence of the problem into a sequence of
quasi-minimal sets which converges for the local Hausdorff distance by using a Federer-Flemming-like
polyhedric approximation theorem by uniformly shaped polyhedrons. We show that the limit set is minimal,
and in some cases this can be enough to control the topology. We give some examples of topological
constraint for which it is possible to provide a complete existence result using this method.
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