10 février 2020

Andreas Juhl ( Humboldt-Universität Berlin)
Singular Yamabe problem, residue families and conformal hypersurface invariants

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Lieu : IMO ; salle 3L8.

Résumé : We describe recent progress on constructions of natural conformally invariant differential operators which are associated to hypersurfaces in Riemannian manifolds. The constructions rest on the solution of a singular version of the Yamabe problem. We outline two basic approaches. The first rests on conformal tractor calculus (Gover-Waldron) and the second generalizes the notion of residue families (introduced by the author) which involves the Feffermann-Graham Poincaré-Einstein metric. We prove the equivalence of both methods. Both constructions are curved analogs of symmetry breaking operators in representation theory (Kobayashi). Among many things, this naturally leads to a notion of extrinsic Q-curvature which generalizes Branson’s Q-curvature. The presentation will describe work of Gover-Waldron, Graham, Juhl-Orsted and others.

Singular Yamabe problem, residue families and conformal hypersurface invariants  Version PDF

Burglind Jöricke (MPIM)
Fundamental groups, slalom curves and extremal length

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Lieu : IMO ; salle 3L8.

Résumé : We define the extremal length of elements of the fundamental group of the twice punctured complex plane and give effective estimates for this invariant. The main motivation comes from 3-braid invariants and their application, for instance to effective finiteness theorems in the spirit of the Geometric Shafarevich Conjecture over Riemann surfaces of second kind.

Fundamental groups, slalom curves and extremal length  Version PDF