Quasiconformal geometry and removable sets for conformal mappings

Abstract:

We are interested in constructing metric spaces by studying lower semicontinuous weights in the plane. Given such a weight, one can define a length distance as in the classical Riemannian setting and study the regularity of the space. We are interested in locally bounded weights that vanish on a Cantor set in the plane, in which case the construction always defines a length distance. We want to understand when the constructed length space admits a quasiconformal parametrization by a Riemannian surface. We provide a sufficient condition on the "allowed" Cantor sets: A compact set is removable for conformal mappings if conformal maps defined on its complement are restrictions of Möbius transformations. Such sets are always Cantor sets and every weight vanishing on it yields a length space admitting a quasiconformal parametrization by a Riemannian surface. We also discuss a suitable converse of this result.

The talk is based on the joint work "Quasiconformal geometry and removable sets for conformal mappings" (to appear in J. Anal. Math.) with Matthew Romney.

Il seminario si terrà in modalità mista, sia in presenza che da remoto. Chi fosse interessato a partecipare in presenza deve scrive a classi@sns.it entro le 14:00 di Mercoledì 2 febbraio 2022.

Link per collegarsi da remoto: https://teams.microsoft.com/l/meetup-join/19%3ameeting_NWEzZThhZDUtZTIzZS00NGRkLWFkZGMtNzU2YmU1OTkwNzNk%40thread.v2/0?context=%7b%22Tid%22%3a%2260240b54-f639-46a1-bf85-a1aba95550fe%22%2c%22Oid%22%3a%223fbb5f06-20de-4c87-8565-7e08a4149047%22%7d