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Flippable tilings of constant curvature surfaces. (English) Zbl 1296.52011

The authors introduce very special tilings of surfaces which they call flippable tilings. Consider tilings of a constant curvature surface by ‘black’ and ‘white’ faces which are convex polygons. Any oriented edge \(e\) of the tiling is a segment adjacent to a black face and a white face on its right side and similarily on its left side. The lengths of the intersections with \(e\) of the two black faces (respectively the two white faces) are equal. In a right flippable tiling the black face is forward on the right-hand side of \(e\) and backward on the left-hand side. In a left flippable tiling, for each edge \(e\), the black face is forward on the left-hand side of \(e\) and backward on the right-hand side.
For a right flippable tiling the operation called a ‘flip’ consists in pushing all black faces forward on the left-hand side and backward on the right-hand side and yields a left flippable tiling. Similarily a flip of a left flippable tiling is defined. Thus pairs of tilings of a surface, where one tiling is obtained from the other by a flip, are objects under study.
The authors prove some existence theorems for the sphere and hyperbolic surfaces. The space of flippable tilings is studied globally both in the spherical and hyperbolic cases. Also more specific results are obtained on the parametrization of the space of flippable tilings for which the areas of black faces are fixed.
Proofs of the results are based on the geometry of polyhedral surfaces in 3-dimensional spaces modelled either on the sphere or on the anti-de Sitter space.
One of the authors’ motivations for this work is the investigation of a polyhedral version of earthquakes.

MSC:

52B70 Polyhedral manifolds
52A15 Convex sets in \(3\) dimensions (including convex surfaces)
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
86A17 Global dynamics, earthquake problems (MSC2010)
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Full Text: arXiv Euclid

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[1] \beginbarticle \bauthor\binitsR. \bsnmAiyama, \bauthor\binitsK. \bsnmAkutagawa and \bauthor\binitsT. \bsnmWan, \batitleMinimal maps between the hyperbolic discs and generalized Gauss maps of maximal surfaces in the anti-de Sitter \(3\)-space, \bjtitleTohoku Math. J. (2) \bvolume52 (\byear2000), no. \bissue3, page 415-\blpage429. \endbarticle \endbibitem · Zbl 0978.53106 · doi:10.2748/tmj/1178207821
[2] \beginbarticle \bauthor\binitsL. \bsnmAndersson, \bauthor\binitsT. \bsnmBarbot, \bauthor\binitsR. \bsnmBenedetti, \bauthor\binitsF. \bsnmBonsante, \bauthor\binitsW. \bsnmGoldman, \bauthor\binitsF. \bsnmLabourie, \bauthor\binitsK. \bsnmScannell and \bauthor\binitsJ.-M. \bsnmSchlenker, \batitleNotes on: “Lorentz spacetimes of constant curvature”, \bcomment[Geom. Dedicata 126 (2007), 3-45; MR 2328921] by G. Mess, \bjtitleGeom. Dedicata \bvolume126 (\byear2007), page 47-\blpage70. \endbarticle \endbibitem · Zbl 1126.53042 · doi:10.1007/s10711-007-9164-6
[3] \beginbbook \bauthor\binitsA. D. \bsnmAlexandrov, \bbtitleConvex polyhedra, \bsertitleSpringer Monographs in Mathematics, \bpublisherSpringer, \blocationBerlin, \byear2005. \bcommentTranslated from the 1950 Russian edition by N. S. Dairbekov, S. S. Kutateladze and A. B. Sossinsky, with comments and bibliography by V. A. Zalgaller and appendices by L. A. Shor and Yu. A. Volkov. \endbbook \endbibitem
[4] \beginbarticle \bauthor\binitsT. \bsnmBarbot, \bauthor\binitsF. and \bauthor\binitsA. \bsnmZeghib, \batitleConstant mean curvature foliations of globally hyperbolic spacetimes locally modelled on \(\mathrm{AdS}_3\), \bjtitleGeom. Dedicata \bvolume126 (\byear2007), page 71-\blpage129. \endbarticle \endbibitem · Zbl 1255.83118 · doi:10.1007/s10711-005-6560-7
[5] \beginbotherref \oauthor\binitsJ. \bsnmBertrand, Prescription of Gauss curvature using optimal mass transport , preprint, 2010. \endbotherref \endbibitem
[6] \beginbarticle \bauthor\binitsF. \bsnmBonsante, \bauthor\binitsK. \bsnmKrasnov and \bauthor\binitsJ.-M. \bsnmSchlenker, \batitleMulti-black holes and earthquakes on Riemann surfaces with boundaries, \bjtitleInt. Math. Res. Not. IMRN \bvolume3 (\byear2011), page 487-\blpage552. \endbarticle \endbibitem · Zbl 1208.30041 · doi:10.1093/imrn/rnq070
[7] \beginbarticle \bauthor\binitsF. \bsnmBonsante and \bauthor\binitsJ.-M. \bsnmSchlenker, \batitleAdS manifolds with particles and earthquakes on singular surfaces, \bjtitleGeom. Funct. Anal. \bvolume19 (\byear2009), no. \bissue1, page 41-\blpage82. \endbarticle \endbibitem · Zbl 1178.32009 · doi:10.1007/s00039-009-0716-9
[8] \beginbarticle \bauthor\binitsF. \bsnmBonsante and \bauthor\binitsJ.-M. \bsnmSchlenker, \batitleMaximal surfaces and the universal Teichmüller space, \bjtitleInvent. Math. \bvolume182 (\byear2010), no. \bissue2, page 279-\blpage333. \endbarticle \endbibitem · Zbl 1222.53063 · doi:10.1007/s00222-010-0263-x
[9] \beginbarticle \bauthor\binitsF. \bsnmBonsante and \bauthor\binitsJ.-M. \bsnmSchlenker, \batitleFixed points of compositions of earthquakes, \bjtitleDuke Math. J. \bvolume161 (\byear2012), no. \bissue6, page 1011-\blpage1054. \endbarticle \endbibitem · Zbl 1244.32007 · doi:10.1215/00127094-1548434
[10] \beginbarticle \bauthor\binitsY. \bsnmCho, \batitleTrigonometry in extended hyperbolic space and extended de Sitter space, \bjtitleBull. Korean Math. Soc. \bvolume46 (\byear2009), no. \bissue6, page 1099-\blpage1133. \endbarticle \endbibitem · Zbl 1194.51022 · doi:10.4134/BKMS.2009.46.6.1099
[11] \beginbarticle \bauthor\binitsM. P. \bparticledo \bsnmCarmo and \bauthor\binitsF. W. \bsnmWarner, \batitleRigidity and convexity of hypersurfaces in spheres, \bjtitleJ. Differential Geometry \bvolume4 (\byear1970), page 133-\blpage144. \endbarticle \endbibitem
[12] \beginbarticle \bauthor\binitsF. \bsnmFillastre, \batitlePolyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces, \bjtitleAnn. Inst. Fourier (Grenoble) \bvolume57 (\byear2007), no. \bissue1, page 163-\blpage195. \endbarticle \endbibitem · Zbl 1123.53033 · doi:10.5802/aif.2255
[13] \beginbarticle \bauthor\binitsF. \bsnmFillastre, \batitleFuchsian polyhedra in Lorentzian space-forms, \bjtitleMath. Ann. \bvolume350 (\byear2011), no. \bissue2, page 417-\blpage453. \endbarticle \endbibitem · Zbl 1229.52017 · doi:10.1007/s00208-010-0563-x
[14] \beginbarticle \bauthor\binitsF. \bsnmFillastre, \batitleFuchsian convex bodies: Basics of Brunn-Minkowski theory, \bjtitleGeom. Funct. Anal. \bvolume23 (\byear2013), no. \bissue1, page 295-\blpage333. \endbarticle \endbibitem · Zbl 1271.52009 · doi:10.1007/s00039-012-0205-4
[15] \beginbotherref \oauthor\binitsF. , Pavages flippés euclidiens , preprint. \endbotherref \endbibitem
[16] \beginbarticle \bauthor\binitsC. W. \bsnmHo, \batitleA note on proper maps, \bjtitleProc. Amer. Math. Soc. \bvolume51 (\byear1975), page 237-\blpage241. \endbarticle \endbibitem · doi:10.1090/S0002-9939-1975-0370471-3
[17] \beginbotherref \oauthor\binitsI. \bsnmIskhakov, On hyperbolic surfaces tessellations and equivariant spacelike convex polyhedral surfaces in Minkowski space , Ph.D. thesis, Ohio State University, 2000. \endbotherref \endbibitem
[18] \beginbbook \bauthor\binitsJ. \bsnmJost, \bbtitleCompact Riemann surfaces, An introduction to contemporary mathematics, \bedition3rd ed., \bpublisherSpringer, \blocationBerlin, \byear2006. \endbbook \endbibitem · Zbl 1125.30033 · doi:10.1007/978-3-540-33067-7
[19] \beginbarticle \bauthor\binitsK. \bsnmKrasnov and \bauthor\binitsJ.-M. \bsnmSchlenker, \batitleMinimal surfaces and particles in \(3\)-manifolds, \bjtitleGeom. Dedicata \bvolume126 (\byear2007), page 187-\blpage254. \endbarticle \endbibitem · Zbl 1126.53037 · doi:10.1007/s10711-007-9132-1
[20] \beginbarticle \bauthor\binitsF. \bsnmLuo and \bauthor\binitsG. \bsnmTian, \batitleLiouville equation and spherical convex polytopes, \bjtitleProc. Amer. Math. Soc. \bvolume116 (\byear1992), no. \bissue4, page 1119-\blpage1129. \endbarticle \endbibitem · Zbl 0806.53012 · doi:10.2307/2159498
[21] \beginbarticle \bauthor\binitsG. \bsnmMess, \batitleLorentz spacetimes of constant curvature, \bjtitleGeom. Dedicata \bvolume126 (\byear2007), page 3-\blpage45. \endbarticle \endbibitem · Zbl 1206.83117 · doi:10.1007/s10711-007-9155-7
[22] \beginbotherref \oauthor\binitsI. \bsnmPak, Lectures on discrete and polyhedral geometry , to be published, preliminary version available at author’s web page at www.math.ucla.edu/ pak/geompol8.pdf. \endbotherref \endbibitem
[23] \beginbbook \bauthor\binitsJ. \bsnmRichter-Gebert, \bbtitleRealization spaces of polytopes, \bsertitleLecture Notes in Mathematics, vol. \bseriesno1643, \bpublisherSpringer, \blocationBerlin, \byear1996. \endbbook \endbibitem
[24] \beginbarticle \bauthor\binitsI. \bsnmRivin and \bauthor\binitsC. D. \bsnmHodgson, \batitleA characterization of compact convex polyhedra in hyperbolic \(3\)-space, \bjtitleInvent. Math. \bvolume111 (\byear1993), no. \bissue1, page 77-\blpage111. \endbarticle \endbibitem · Zbl 0784.52013 · doi:10.1007/BF01231281
[25] \beginbotherref \oauthor\binitsJ.-M. \bsnmSchlenker, Hyperbolic manifolds with polyhedral boundary , available at \arxivurl
[26] \beginbarticle \bauthor\binitsJ.-M. \bsnmSchlenker, sur les polyèdres hyperboliques convexes, \bjtitleJ. Differential Geom. \bvolume48 (\byear1998), no. \bissue2, page 323-\blpage405. \endbarticle \endbibitem
[27] \beginbchapter \bauthor\binitsJ.-M. \bsnmSchlenker, \bctitleDes immersions isométriques de surfaces aux variétés hyperboliques à bord convexe, de Théorie Spectrale et Géométrie, vol. 21, Année 2002-2003, . Théor. Spectr. Géom., vol. \bseriesno21, \bpublisherUniv. Grenoble I, \blocationSaint-Martin-d’Hères, \byear2003, pp. page 165-\blpage216. \endbchapter \endbibitem
[28] \beginbarticle \bauthor\binitsJ.-M. \bsnmSchlenker, \batitleSmall deformations of polygons and polyhedra, \bjtitleTrans. Amer. Math. Soc. \bvolume359 (\byear2007), no. \bissue5, page 2155-\blpage2189. \endbarticle \endbibitem · Zbl 1126.53041 · doi:10.1090/S0002-9947-06-04172-9
[29] \beginbotherref \oauthor\binitsW. P. \bsnmThurston, The geometry and topology of three-manifolds , recent version of the 1980 notes, available at library.msri.org/books/gt3m/, 1997. \endbotherref \endbibitem
[30] \beginbchapter \bauthor\binitsM. \bsnmTroyanov, \bctitleMetrics of constant curvature on a sphere with two conical singularities, \bbtitleDifferential geometry (, \bconfdate1988), \bsertitleLecture Notes in Math., vol. \bseriesno1410, \bpublisherSpringer, \blocationBerlin, \byear1989, pp. page 296-\blpage306. \endbchapter \endbibitem · doi:10.1007/BFb0086431
[31] \beginbarticle \bauthor\binitsM. \bsnmTroyanov, \batitlePrescribing curvature on compact surfaces with conical singularities, \bjtitleTrans. Amer. Math. Soc. \bvolume324 (\byear1991), no. \bissue2, page 793-\blpage821. \endbarticle \endbibitem · Zbl 0724.53023 · doi:10.2307/2001742
[32] \beginbchapter \bauthor\binitsM. \bsnmTroyanov, \bctitleSurfaces riemanniennes à singularités simples, \bbtitleDifferential geometry: Geometry in mathematical physics and related topics (\bconfnameLos Angeles, \bconflocationCA, \bconfdate1990), \bsertitleProc. Sympos. Pure Math., vol. \bseriesno54, \bpublisherAmer. Math. Soc., \blocationProvidence, RI, \byear1993, pp. page 619-\blpage628. \endbchapter \endbibitem · doi:10.1090/pspum/054.2/1216569
[33] \beginbarticle \bauthor\binitsJ. \bparticleVan \bsnmHeijenoort, \batitleOn locally convex manifolds, \bjtitleComm. Pure Appl. Math. \bvolume5 (\byear1952), page 223-\blpage242. \endbarticle \endbibitem · Zbl 0049.12201 · doi:10.1002/cpa.3160050302
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