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Continuity of the bending map

Cyril Lecuire (2008)

Annales de la faculté des sciences de Toulouse Mathématiques

The bending map of a hyperbolic 3 -manifold maps a convex cocompact hyperbolic metric on a 3 -manifold with boundary to its bending measured geodesic lamination. As proved in [KeS] and [KaT], this map is continuous. In the present paper we study the extension of this map to the space of geometrically finite hyperbolic metrics. We introduce a relationship on the space of measured geodesic laminations and show that the quotient map obtained from the bending map is continuous.

Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocations dynamics

Francesca Da Lio, N. Forcadel, Régis Monneau (2008)

Journal of the European Mathematical Society

We prove the convergence at a large scale of a non-local first order equation to an anisotropic mean curvature motion. The equation is an eikonal-type equation with a velocity depending in a non-local way on the solution itself, which arises in the theory of dislocation dynamics. We show that if an anisotropic mean curvature motion is approximated by equations of this type then it is always of variational type, whereas the converse is true only in dimension two.

Convergence of Bergman geodesics on CP 1

Jian Song, Steve Zelditch (2007)

Annales de l’institut Fourier

The space of Kähler metrics in a fixed Kähler class on a projective Kähler manifold X is an infinite dimensional symmetric space whose geodesics ω t are solutions of a homogeneous complex Monge-Ampère equation in A × X , where A is an annulus. Phong-Sturm have proven that the Monge-Ampère geodesic of Kähler potentials ϕ ( t , z ) of ω t may be approximated in a weak C 0 sense by geodesics ϕ N ( t , z ) of the finite dimensional symmetric space of Bergman metrics of height N . In this article we prove that ϕ N ( t , z ) ϕ ( t , z ) in C 2 ( [ 0 , 1 ] × X ) in the case of...

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