Lokale Starrheit 3-dimensionaler Kegelmannigfaltigkeiten

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Lokale Starrheit 3-dimensionaler Kegelmannigfaltigkeiten

Local rigidity of 3-dimensional cone-manifolds

Weiß, Hartmut
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URI: http://nbn-resolving.de/urn:nbn:de:bsz:21-opus-6662
http://hdl.handle.net/10900/48427
http://nbn-resolving.org/urn:nbn:de:bsz:21-dspace-484279
http://nbn-resolving.org/urn:nbn:de:bsz:21-dspace-484278
Dokumentart: PhDThesis
Date: 2002
Language: German
Faculty: 7 Mathematisch-Naturwissenschaftliche Fakultät
Department: Sonstige - Mathematik und Physik
Advisor: Leeb, Bernhard
Day of Oral Examination: 2002-12-19
DDC Classifikation: 510 - Mathematics
Keywords: Niederdimensionale Topologie , Mannigfaltigkeit / Dimension 3 , Einsteinsche Mannigfaltigkeit , Hyperbolische Mannigfaltigkeit
Other Keywords: Geometrisierung von 3-Mannigfaltigkeiten , Deformationen von Kegelmannigfaltigkeitsstrukturen
Geometric topology , Geometrization of 3-manifolds , Deformations of cone-manifold structures
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Inhaltszusammenfassung:

We investigate local rigidity of 3-dimensional cone-manifolds with cone-angles not larger than $pi$. Under this cone-angle restriction the singular locus is a trivalent graph. We obtain local rigidity in the hyperbolic and the spherical case. From a technical point of view the main result is a vanishing theorem for $L^2$-cohomology of the smooth part of the cone-manifold with coefficients in the flat vectorbundle of infinitesimal isometries. From this local rigidity is deduced by an analysis of the variety of representations.

Abstract:

We investigate local rigidity of 3-dimensional cone-manifolds with cone-angles not larger than $pi$. Under this cone-angle restriction the singular locus is a trivalent graph. We obtain local rigidity in the hyperbolic and the spherical case. From a technical point of view the main result is a vanishing theorem for $L^2$-cohomology of the smooth part of the cone-manifold with coefficients in the flat vectorbundle of infinitesimal isometries. From this local rigidity is deduced by an analysis of the variety of representations.

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