The purpose of this note is to discuss examples of geometric transition from hyperbolic structures to half-pipe and Anti–de Sitter structures in dimensions two, three and four. As a warm-up, explicit examples of transition to Euclidean and spherical structures are presented. No new results appear here; nor an exhaustive treatment is aimed. 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].
@article{TSG_2017-2019__35__163_0, author = {Andrea Seppi}, title = {Examples of geometric transition in low dimensions}, journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie}, pages = {163--196}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {35}, year = {2017-2019}, doi = {10.5802/tsg.368}, language = {en}, url = {https://proceedings.centre-mersenne.org/articles/10.5802/tsg.368/} }
TY - JOUR AU - Andrea Seppi TI - Examples of geometric transition in low dimensions JO - Séminaire de théorie spectrale et géométrie PY - 2017-2019 SP - 163 EP - 196 VL - 35 PB - Institut Fourier PP - Grenoble UR - https://proceedings.centre-mersenne.org/articles/10.5802/tsg.368/ DO - 10.5802/tsg.368 LA - en ID - TSG_2017-2019__35__163_0 ER -
%0 Journal Article %A Andrea Seppi %T Examples of geometric transition in low dimensions %J Séminaire de théorie spectrale et géométrie %D 2017-2019 %P 163-196 %V 35 %I Institut Fourier %C Grenoble %U https://proceedings.centre-mersenne.org/articles/10.5802/tsg.368/ %R 10.5802/tsg.368 %G en %F TSG_2017-2019__35__163_0
Andrea Seppi. Examples of geometric transition in low dimensions. Séminaire de théorie spectrale et géométrie, Volume 35 (2017-2019), pp. 163-196. doi : 10.5802/tsg.368. https://proceedings.centre-mersenne.org/articles/10.5802/tsg.368/
[1] Norbert A’Campo; Athanase Papadopoulos Sophus Lie and Felix Klein: the Erlangen program and its impact in mathematics and physics, IRMA Lectures in Mathematical and Theoretical Physics, 23, European Mathematical Society, 2015 | MR | Zbl
[2] Thierry Barbot; François Fillastre Quasi-Fuchsian co-Minkowski manifolds (2018) (https://arxiv.org/abs/1801.10429)
[3] Michel Boileau; Bernhard Leeb; Joan Porti Geometrization of 3-dimensional orbifolds, Ann. Math., Volume 162 (2005) no. 1, pp. 195-290 | DOI | MR | Zbl
[4] Daryl Cooper; Jeffrey Danciger; Anna Wienhard Limits of geometries, Trans. Am. Math. Soc., Volume 370 (2018) no. 9, pp. 6585-6627 | DOI | MR | Zbl
[5] Daryl Cooper; Jeffrey Danciger; Anna Wienhard Limits of Geometries, Trans. Am. Math. Soc., Volume 370 (2018) no. 9, pp. 6585-6627 | DOI | MR | Zbl
[6] Daryl Cooper; Craig D. Hodgson; Steven P. Kerckhoff Three-dimensional orbifolds and cone-manifolds, MSJ Memoirs, 5, Mathematical Society of Japan, 2000 (With a postface by Sadayoshi Kojima) | MR | Zbl
[7] Jeffrey Danciger Geometric transition: from hyperbolic to AdS geometry, Ph. D. Thesis, Stanford University, USA (2011) | MR
[8] Jeffrey Danciger A geometric transition from hyperbolic to Anti–de Sitter geometry, Geom. Topol., Volume 17 (2013) no. 5, pp. 3077-3134 | DOI | MR | Zbl
[9] Jeffrey Danciger Ideal triangulations and geometric transitions, J. Topol., Volume 7 (2014) no. 4, pp. 1118-1154 | DOI | MR | Zbl
[10] François Fillastre; Andrea Seppi Spherical, hyperbolic, and other projective geometries: convexity, duality, transitions, Eighteen essays in non-Euclidean geometry (IRMA Lectures in Mathematics and Theoretical Physics), Volume 29, European Mathematical Society, 2019, pp. 321-409 | DOI | MR | Zbl
[11] Damian Heard; Ekaterina Pervova; Carlo Petronio The 191 orientable octahedral manifolds, Exp. Math., Volume 17 (2008) no. 4, pp. 473-486 | DOI | MR | Zbl
[12] Craig D. Hodgson Degeneration and regeneration of hyperbolic structures on three-manifolds (foliations, Dehn surgery), Ph. D. Thesis, Princeton University (1986) | MR
[13] Steven P. Kerckhoff; Peter A. Storm From the hyperbolic 24-cell to the cuboctahedron, Geom. Topol., Volume 14 (2010) no. 3, pp. 1383-1477 | DOI | MR | Zbl
[14] Felix Klein Ueber die sogenannte Nicht-Euklidische Geometrie. (Zweiter Aufsatz), Math. Ann., Volume 6 (1873) no. 2, pp. 112-145 | DOI | MR | Zbl
[15] Bruno Martelli; Stefano Riolo Hyperbolic Dehn filling in dimension four, Geom. Topol., Volume 22 (2018) no. 3, pp. 1647-1716 | DOI | MR | Zbl
[16] Geoffrey Mess Lorentz spacetimes of constant curvature, Geom. Dedicata, Volume 126 (2007), pp. 3-45 | DOI | MR | Zbl
[17] Joan Porti Regenerating hyperbolic and spherical cone structures from Euclidean ones, Topology, Volume 37 (1998) no. 2, pp. 365-392 | DOI | MR | Zbl
[18] Joan Porti Regenerating hyperbolic cone 3-manifolds from dimension 2, Ann. Inst. Fourier, Volume 63 (2013) no. 5, pp. 1971-2015 | DOI | Numdam | MR | Zbl
[19] Stefano Riolo; Andrea Seppi Geometric transition from hyperbolic to Anti-de Sitter structures in dimension four (2019) (https://arxiv.org/abs/1908.05112,to be published in Annali della Scuola Normale Superiore, Classe di Scienze)
[20] Stefano Riolo; Andrea Seppi Character varieties of a transitioning Coxeter 4-orbifold (2020) (https://arxiv.org/abs/2006.15847)
[21] Caroline Series Limits of quasi-Fuchsian groups with small bending, Duke Math. J., Volume 128 (2005) no. 2, pp. 285-329 | DOI | MR | Zbl
[22] William P. Thurston The geometry and topology of three-manifolds, 1980 (ftp://www.geom.uiuc.edu/priv/levy/incoming/bayer/gtm3.pdf/6a.pdf)
[23] Steve J. Trettel Families of geometries, real algebras, and transitions, Ph. D. Thesis, University of California, USA (2019) | MR
Cited by Sources: