Geometry & Topology

Infinitesimal projective rigidity under Dehn filling

Michael Heusener and Joan Porti

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To a hyperbolic manifold one can associate a canonical projective structure and a fundamental question is whether or not it can be deformed. In particular, the canonical projective structure of a finite volume hyperbolic manifold with cusps might have deformations which are trivial on the cusps.

The aim of this article is to prove that if the canonical projective structure on a cusped hyperbolic manifold M is infinitesimally projectively rigid relative to the cusps, then infinitely many hyperbolic Dehn fillings on M are locally projectively rigid. We analyze in more detail the figure eight knot and the Whitehead link exteriors, for which we can give explicit infinite families of slopes with projectively rigid Dehn fillings.

Article information

Geom. Topol., Volume 15, Number 4 (2011), 2017-2071.

Received: 8 May 2010
Revised: 10 August 2011
Accepted: 13 September 2011
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 53A20: Projective differential geometry 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.)

projective structure variety of representations infinitesimal deformation


Heusener, Michael; Porti, Joan. Infinitesimal projective rigidity under Dehn filling. Geom. Topol. 15 (2011), no. 4, 2017--2071. doi:10.2140/gt.2011.15.2017.

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