We introduce and study some deformations of complete finite-volume hyperbolic
four-manifolds that may be interpreted as four-dimensional analogues of
Thurston's hyperbolic Dehn filling.
We construct in particular an analytic path of complete, finite-volume cone
four-manifolds Mt that interpolates between two hyperbolic four-manifolds
M0 and M1 with the same volume 38π2. The deformation looks
like the familiar hyperbolic Dehn filling paths that occur in dimension three,
where the cone angle of a core simple closed geodesic varies monotonically from
0 to 2π. Here, the singularity of Mt is an immersed geodesic surface
whose cone angles also vary monotonically from 0 to 2π. When a cone angle
tends to 0 a small core surface (a torus or Klein bottle) is drilled
producing a new cusp.
We show that various instances of hyperbolic Dehn fillings may arise,
including one case where a degeneration occurs when the cone angles tend to
2π, like in the famous figure-eight knot complement example.
The construction makes an essential use of a family of four-dimensional
deforming hyperbolic polytopes recently discovered by Kerckhoff and Storm.Comment: 60 pages, 23 figures. Final versio