Deformations of noncompact projective manifolds
In the paper under review the author discusses the existence of strictly convex flexing deformations for noncompact finite-volume 3-manifolds. The author begins by exploring two-bridge knots and links and obtains the following result:NEWLINENEWLINE Let \(M\) be the complement in \(S^{3}\) of \(4_{1}\) (the figure-eight knot), \(5_{2}\), \(6_{1}\), or \(5_{1}^{2}\) (the Whitehead link). Then \(M\) does not admit strictly convex deformations of its complete hyperbolic structure.NEWLINENEWLINE Using some special properties of amphicheiral knots and their representations, the author shows a complementary result:NEWLINENEWLINE Let \(M\) be the complement of a hyperbolic amphicheiral knot, and suppose that \(M\) is infinitesimally projectively rigid relative to the boundary and the longitude is a rigid slope. Then for sufficiently large \(n\), there is a one-dimensional family of strictly convex deformations of the complete hyperbolic structure on \(M(n/0)\). Here \(M(n/0)\) is the orbifold obtained by surgering a solid torus with longitudinal singular locus of cone angle \(2\pi/n\) along the meridian of \(M\).
scientific article; zbMATH DE number 6369092
scientific article