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Elliptic Submanifolds in Hyperbolic Geometry

This project studies the application of analytical tools to the resolution of geometric problems. The geometric problems proposed to be studied arise from hyperbolic geometry and representation theory. More precisely, it is propos... ver más

30/09/2012

CONSORCI CENTRE DE...

147K€

Presupuesto del proyecto: 147K€

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CONSORCI CENTRE DE RECERCA MATEMATICA Otra investigación y desarrollo experimental en ciencias naturales y técnicas otros tipos

TRL 4-5 | 50K€

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Financiación concedida El organismo FP7 notifico la concesión del proyecto el día 2012-09-30 No tenemos la información de la convocatoria

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CONSORCI CENTRE DE RECERCA MATEMATICA Otra investigación y desarrollo experimental en ciencias naturales y técnicas otros tipos

TRL 4-5 | 50K€

Presupuesto del proyecto 147K€

Fecha límite de participación Sin fecha límite de participación.

Descripción del proyecto This project studies the application of analytical tools to the resolution of geometric problems. The geometric problems proposed to be studied arise from hyperbolic geometry and representation theory. More precisely, it is proposed to investigate hyperbolic ends and flat conformal structures (FCSs). The former constitute an important part of the study of hyperbolic manifolds in general, since they arise when the Nielsen Kernel (which is bounded) is removed from a (convex, co-compact) hyperbolic manifold, nonetheless, the structure of the moduli space of hyperbolic ends as well as its compactification remains to be understood. Part of this project is devoted to the study of this problem. The latter arise as a natural geometric structure in the theory of representations. They are intimitely related to hyperbolic ends, but the properties of this identification remain to be fully understood, and part of the project is devoted to the resolution of this problem. This should also yield geometric structures on the moduli space of FCSs. Finally, in the two dimensional case, these structures yield continuous curves inside the moduli space of FCSs, whose geometric properties we propose to investigate. The tools used are mostly immersed submanifolds satisfying elliptic conditions, such as constant curvature. In particular, it is proposed to use a new concept of curvature, developed by the applicant, called special Lagrangian (SL) curvature, which captures the important convexity properties of Gaussian curvature whilst overcoming its technical limitations. In order to fully realise the potential of SL curvature as a geometric tool, various properties (especially compactness) remain to be fully understood, to which the final part of the project is devoted.

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