Title : The uniqueness theorem for Euclidean two-bridge knot cone-manifolds
Speaker : Rusian N. Shmatkov (Novosibirsk State University)
Abstract :

A {\em 3-dimensional Euclidean \cm\ } is a metric space obtained as the quotient space of a disjoint union of a collection of geodesic 3-simplices in the 3-dimensional Euclidean space ${\Bbb E}^3$ by an isometric pairing of faces in such a combinatorial fashion that the underlying topological space is a manifold. {\em Hyperbolic} and {\em spherical cone-manifolds} are defined similarly. Such a space possesses a Riemannian metric of constant sectional curvature on the union of the top-dimensional cells and the dimension-2 cells. On each dimension-1 cell, the structure is completely described by an angle, which is the sum of the dihedral angles around all of the dimension-1 simplicial faces which are identified to give the cell. The {\em singular set } of a cone-manifold is the closure of all the dimension-1 cells for which this angle, called the {\em cone angle}, is not $2\pi$ (the Riemannian metric may be extended smoothly over all cells whose angle is $2\pi$). Let $(\Ss^3, (p/q)_{\al})$ be a cone-manifold whose underlying space is the 3-sphere and singular set is a 2-bridge knot $p/q$ with a cone angle $\al$. The main result of this report is the following theorem. \vspace{\baselineskip} {\bf The Uniqueness Theorem.} {\em If a cone-manifold $(\Ss^3, (p/q)_{\al})$ admits the Euclidean structure, then there is unique cone angle $0<\al_e<\pi$ such that the cone-manifold $(\Ss^3, (p/q)_{\al_e})$ is Euclidean.} \vspace{\baselineskip} This theorem is proved by using results of S.~Kojima \cite{Kojima}, C.D.~Hodgson, S.P.~Kerckhoff \cite{Kerckhoff}, G.D.~Mostow \cite{Mostow} and J.~Porti \cite{Porti}. \begin{thebibliography}{99} \bibitem{Kojima} Kojima S. {\em Deformations of hyperbolic 3-cone-manifolds} // J. Differential Geom. --- 1998. --- V. 49. --- P. 469--516. \bibitem{Kerckhoff} Hodgson C.D., Kerckhoff S.P. {\em Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery} // J. Differential Geom. --- 1998. --- V. 48. --- P. 1--59. \bibitem{Mostow} Mostow G.D. {\em Quasi-conformal mappings in $n$-space and the rigidity of hyperbolic space forms} // Inst. Hautes \'Etudes. Sci. Publ. Math. --- 1968. --- V. 34. --- P. 53--104. \bibitem{Porti} Porti J. {\em Regenerating hyperbolic and spherical cone structures from Euclidean ones} // Topology. --- 1998. --- V. 37, N. 2. --- P. 365--392. \end{thebibliography}

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