Analysis and Geometry of Low-dimensional Manifolds


CALL: 2016

DOMAIN: MT - Mathematics


LAST NAME: Schlenker



HOST INSTITUTION: University of Luxembourg

KEYWORDS: hyperbolic geometry, anti-de sitter geometry, teichmuller theory, geometric structures, manifolds with particles

START: 2017-09-01

END: 2020-08-31


Submitted Abstract

We will study several related but distinct aspects of geometric structures on low-dimensionalmanifolds, focussing mainly on hyperbolic and AdS structures on 3-dimensional manifolds and on parabolic geometric structures and their automorphism groups. The project will blend together geometric, analytic and algebraic methods to achieve new results in those active areas of current research.The most studied type of geometric structures are hyperbolic metrics. In the first part of the project we will expand our understanding of the renormalized volume of quasifuchsian manifolds, as well as develop new aspects of their geometry suggested by a deep analogy between the renormalized volume and data “at infinity” on one hand, and the volume of the convex core and data on its boundary, on the other hand.Still in the area of hyperbolic geometry, we intend to study a new type of map between a hyperbolic surface and a 3-dimensional manifold (or more generally a negatively curved Riemannian manifold) that minimizes the 1-Schatten norm of the differential. We believe that those maps, which are motivated by our recent work on applications of minimal Lagrangian diffeomorphisms in Teichmüller theory, share some interesting properties of both harmonic maps and pleated surfaces.In a third line of investigations, we will study the “universal” analogs of well-known conjectures of Thurston (resp. Mess) on the induced metric and measured bending laminations on the convex cores of quasifuchsian (resp. globally hyperbolic anti-de Sitter) manifolds.Part 4 and 5 of the project are concerned with hyperbolic (resp. anti-de Sitter) manifolds with “particles”, that is, cone singularities along (time-like) lines, with cone angles less than p. In both cases we intend to extend to this setting a number of key results known for non-singular manifolds.Finally the last part of the project will study parabolic geometric structures on low-dimensional manifolds, and strive to classify those with a large group of automorphisms, therefore contributing to a full understanding of the “generalized Lichnerowicz conjecture”.

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