Vlad Moraru and Vincent Feuvrier
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 |
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.
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