Geometric transition from hyperbolic to anti-de Sitter structures in dimension four

Abstract

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Examples of geometric transition in low dimensions

“…In [19] it is proved that, when rescaled by η |t| , the polytope P(t) converges to a half-pipe polytope, for which the walls of the form X and i − are degenerate, while those of the form i + are non-degenerate. This is used to produce a geometric transition hyperbolic/half-pipe/Anti-de Sitter on four-dimensional manifolds.…”

“…In [8] (see also [9]) the existence of phenomena of regeneration from half-pipe structures in dimension three was proven, under the hypothesis of a cohomological condition in the spirit of [17]. The recent paper [19] provided the first examples of geometric transition in dimension four, on a certain class of cusped, finite-volume manifolds. See also the companion paper [20].…”

“…The purpose of this note is to describe rather explicitly several examples of geometric transition from hyperbolic to half-pipe and Anti-de Sitter geometry, first in dimension two and three, which is done in Section 3, and then to outline the four-dimensional examples of [19], which is the content of Section 4. But before that, in Section 2 certain examples of transition from hyperbolic to Euclidean and spherical geometry are provided as a sort of warm-up, again in dimension two and three.…”

“…Although mostly elementary, we provide some of the details of these examples in a rather precise fashion, with the purpose of highlighting the methods involved. As already stated, in Section 4 we give a brief outline of the four-dimensional construction of the recent work [19], which is based on a deforming polytope introduced in [13] and [15], and we rely on the previous lower dimensional examples to help the four-dimensional intuition.…”

“…On the other hand, details of some elementary computations are provided to explain certain techniques involved. This note, and in particular the last section, can also serve as an introduction to the ideas behind the four-dimensional construction of [19].…”