Metristen pintojen uniformisaatio
Päärahoittaja
Päärahoittajan myöntämä tuki (€)
- 25 000,00
Hankkeen aikataulu
Hankkeen aloituspäivämäärä: 01.12.2020
Hankkeen päättymispäivämäärä: 30.11.2021
Tiivistelmä
During the last two years, the applicant proved a uniformization result for metric surfaces - metric spaces with locally finite Hausdorff 2-measure that are homeomorphic to a connected 2-manifold. More precisely, we studied a subclass of metric surfaces that are 'locally reciprocal'. By definition, they are the metric surfaces that can be covered by quasiconformal images of planar domains. We use the so-called geometric definition of quasiconformality.
We are interested in the following two classes of questions. Can we sew two locally reciprocal surfaces X and Y to
obtain a locally reciprocal surface Z? The motivation for this question is to be able to construct locally reciprocal surfaces from smaller pieces.
Is a locally reciprocal metric surface X conformally equivalent to a metric surface Z that has better geometric properties? This problem is not that hard when the distance on X is sufficiently regular, in which case we already
know that we can construct Z that is conformally equivalent to X and biLipschitz equivalent to a complete
Riemannian surface of constant curvature. The problem is harder in general and we are interested in solving it or
finding an example of a locally reciprocal X that is not conformally equivalent to a better metric surface.
We are interested in the following two classes of questions. Can we sew two locally reciprocal surfaces X and Y to
obtain a locally reciprocal surface Z? The motivation for this question is to be able to construct locally reciprocal surfaces from smaller pieces.
Is a locally reciprocal metric surface X conformally equivalent to a metric surface Z that has better geometric properties? This problem is not that hard when the distance on X is sufficiently regular, in which case we already
know that we can construct Z that is conformally equivalent to X and biLipschitz equivalent to a complete
Riemannian surface of constant curvature. The problem is harder in general and we are interested in solving it or
finding an example of a locally reciprocal X that is not conformally equivalent to a better metric surface.