Rational maps as Schwarzian primitives

Abstract We examine when a meromorphic quadratic differential ϕ with prescribed poles is the Schwarzian derivative of a rational map. We give a necessary and sufficient condition: In the Laurent series of ϕ around each pole c, the most singular term should take the form (1 − d2)/(2(z − c)2), where d...
Ausführliche Beschreibung

Gespeichert in:

Autor*in:	Cui, GuiZhen [verfasserIn] Gao, Yan [verfasserIn] Rugh, Hans Henrik [verfasserIn] Tan, Lei [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2016

Schlagwörter:	Schwarzian derivatives rational maps critical points meromorphic quadratic differentials

Übergeordnetes Werk:	Enthalten in: Science in China - Asheville, NC : Science in China Press, 1995, 59(2016), 7 vom: 29. März, Seite 1267-1284
Übergeordnetes Werk:	volume:59 ; year:2016 ; number:7 ; day:29 ; month:03 ; pages:1267-1284

Links:	Volltext

DOI / URN:	10.1007/s11425-016-5140-7

Katalog-ID:	SPR019143370

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520			\|a Abstract We examine when a meromorphic quadratic differential ϕ with prescribed poles is the Schwarzian derivative of a rational map. We give a necessary and sufficient condition: In the Laurent series of ϕ around each pole c, the most singular term should take the form (1 − d2)/(2(z − c)2), where d is an integer, and then a certain determinant in the next d coefficients should vanish. This condition can be optimized by neglecting some information on one of the poles (i.e., by only requiring it to be a double pole). The case d = 2 was treated by Eremenko (2012). We show that a geometric interpretation of our condition is that the complex projective structure induced by ϕ outside the poles has a trivial holonomy group. This statement was suggested to us by Thurston in a private communication. Our work is related to the problem of finding a rational map f with a prescribed set of critical points, since the critical points of f are precisely the poles of its Schwarzian derivative. Finally, we study the pole-dependency of these Schwarzian derivatives. We show that, in the cubic case with simple critical points, an analytic dependency fails precisely when the poles are displaced at the vertices of a regular ideal tetrahedron of the hyperbolic 3-ball.
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allfields	10.1007/s11425-016-5140-7 doi (DE-627)SPR019143370 (SPR)s11425-016-5140-7-e DE-627 ger DE-627 rakwb eng 510 530 520 ASE 30.00 bkl 31.00 bkl Cui, GuiZhen verfasserin aut Rational maps as Schwarzian primitives 2016 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract We examine when a meromorphic quadratic differential ϕ with prescribed poles is the Schwarzian derivative of a rational map. We give a necessary and sufficient condition: In the Laurent series of ϕ around each pole c, the most singular term should take the form (1 − d2)/(2(z − c)2), where d is an integer, and then a certain determinant in the next d coefficients should vanish. This condition can be optimized by neglecting some information on one of the poles (i.e., by only requiring it to be a double pole). The case d = 2 was treated by Eremenko (2012). We show that a geometric interpretation of our condition is that the complex projective structure induced by ϕ outside the poles has a trivial holonomy group. This statement was suggested to us by Thurston in a private communication. Our work is related to the problem of finding a rational map f with a prescribed set of critical points, since the critical points of f are precisely the poles of its Schwarzian derivative. Finally, we study the pole-dependency of these Schwarzian derivatives. We show that, in the cubic case with simple critical points, an analytic dependency fails precisely when the poles are displaced at the vertices of a regular ideal tetrahedron of the hyperbolic 3-ball. Schwarzian derivatives (dpeaa)DE-He213 rational maps (dpeaa)DE-He213 critical points (dpeaa)DE-He213 meromorphic quadratic differentials (dpeaa)DE-He213 Gao, Yan verfasserin aut Rugh, Hans Henrik verfasserin aut Tan, Lei verfasserin aut Enthalten in Science in China Asheville, NC : Science in China Press, 1995 59(2016), 7 vom: 29. März, Seite 1267-1284 (DE-627)325695059 (DE-600)2038800-7 1862-2763 nnns volume:59 year:2016 number:7 day:29 month:03 pages:1267-1284 https://dx.doi.org/10.1007/s11425-016-5140-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-AST SSG-OPC-ASE GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_152 GBV_ILN_161 GBV_ILN_171 GBV_ILN_187 GBV_ILN_224 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 30.00 ASE 31.00 ASE AR 59 2016 7 29 03 1267-1284
spelling	10.1007/s11425-016-5140-7 doi (DE-627)SPR019143370 (SPR)s11425-016-5140-7-e DE-627 ger DE-627 rakwb eng 510 530 520 ASE 30.00 bkl 31.00 bkl Cui, GuiZhen verfasserin aut Rational maps as Schwarzian primitives 2016 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract We examine when a meromorphic quadratic differential ϕ with prescribed poles is the Schwarzian derivative of a rational map. We give a necessary and sufficient condition: In the Laurent series of ϕ around each pole c, the most singular term should take the form (1 − d2)/(2(z − c)2), where d is an integer, and then a certain determinant in the next d coefficients should vanish. This condition can be optimized by neglecting some information on one of the poles (i.e., by only requiring it to be a double pole). The case d = 2 was treated by Eremenko (2012). We show that a geometric interpretation of our condition is that the complex projective structure induced by ϕ outside the poles has a trivial holonomy group. This statement was suggested to us by Thurston in a private communication. Our work is related to the problem of finding a rational map f with a prescribed set of critical points, since the critical points of f are precisely the poles of its Schwarzian derivative. Finally, we study the pole-dependency of these Schwarzian derivatives. We show that, in the cubic case with simple critical points, an analytic dependency fails precisely when the poles are displaced at the vertices of a regular ideal tetrahedron of the hyperbolic 3-ball. Schwarzian derivatives (dpeaa)DE-He213 rational maps (dpeaa)DE-He213 critical points (dpeaa)DE-He213 meromorphic quadratic differentials (dpeaa)DE-He213 Gao, Yan verfasserin aut Rugh, Hans Henrik verfasserin aut Tan, Lei verfasserin aut Enthalten in Science in China Asheville, NC : Science in China Press, 1995 59(2016), 7 vom: 29. März, Seite 1267-1284 (DE-627)325695059 (DE-600)2038800-7 1862-2763 nnns volume:59 year:2016 number:7 day:29 month:03 pages:1267-1284 https://dx.doi.org/10.1007/s11425-016-5140-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-AST SSG-OPC-ASE GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_152 GBV_ILN_161 GBV_ILN_171 GBV_ILN_187 GBV_ILN_224 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 30.00 ASE 31.00 ASE AR 59 2016 7 29 03 1267-1284
allfields_unstemmed	10.1007/s11425-016-5140-7 doi (DE-627)SPR019143370 (SPR)s11425-016-5140-7-e DE-627 ger DE-627 rakwb eng 510 530 520 ASE 30.00 bkl 31.00 bkl Cui, GuiZhen verfasserin aut Rational maps as Schwarzian primitives 2016 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract We examine when a meromorphic quadratic differential ϕ with prescribed poles is the Schwarzian derivative of a rational map. We give a necessary and sufficient condition: In the Laurent series of ϕ around each pole c, the most singular term should take the form (1 − d2)/(2(z − c)2), where d is an integer, and then a certain determinant in the next d coefficients should vanish. This condition can be optimized by neglecting some information on one of the poles (i.e., by only requiring it to be a double pole). The case d = 2 was treated by Eremenko (2012). We show that a geometric interpretation of our condition is that the complex projective structure induced by ϕ outside the poles has a trivial holonomy group. This statement was suggested to us by Thurston in a private communication. Our work is related to the problem of finding a rational map f with a prescribed set of critical points, since the critical points of f are precisely the poles of its Schwarzian derivative. Finally, we study the pole-dependency of these Schwarzian derivatives. We show that, in the cubic case with simple critical points, an analytic dependency fails precisely when the poles are displaced at the vertices of a regular ideal tetrahedron of the hyperbolic 3-ball. Schwarzian derivatives (dpeaa)DE-He213 rational maps (dpeaa)DE-He213 critical points (dpeaa)DE-He213 meromorphic quadratic differentials (dpeaa)DE-He213 Gao, Yan verfasserin aut Rugh, Hans Henrik verfasserin aut Tan, Lei verfasserin aut Enthalten in Science in China Asheville, NC : Science in China Press, 1995 59(2016), 7 vom: 29. März, Seite 1267-1284 (DE-627)325695059 (DE-600)2038800-7 1862-2763 nnns volume:59 year:2016 number:7 day:29 month:03 pages:1267-1284 https://dx.doi.org/10.1007/s11425-016-5140-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-AST SSG-OPC-ASE GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_152 GBV_ILN_161 GBV_ILN_171 GBV_ILN_187 GBV_ILN_224 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 30.00 ASE 31.00 ASE AR 59 2016 7 29 03 1267-1284
allfieldsGer	10.1007/s11425-016-5140-7 doi (DE-627)SPR019143370 (SPR)s11425-016-5140-7-e DE-627 ger DE-627 rakwb eng 510 530 520 ASE 30.00 bkl 31.00 bkl Cui, GuiZhen verfasserin aut Rational maps as Schwarzian primitives 2016 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract We examine when a meromorphic quadratic differential ϕ with prescribed poles is the Schwarzian derivative of a rational map. We give a necessary and sufficient condition: In the Laurent series of ϕ around each pole c, the most singular term should take the form (1 − d2)/(2(z − c)2), where d is an integer, and then a certain determinant in the next d coefficients should vanish. This condition can be optimized by neglecting some information on one of the poles (i.e., by only requiring it to be a double pole). The case d = 2 was treated by Eremenko (2012). We show that a geometric interpretation of our condition is that the complex projective structure induced by ϕ outside the poles has a trivial holonomy group. This statement was suggested to us by Thurston in a private communication. Our work is related to the problem of finding a rational map f with a prescribed set of critical points, since the critical points of f are precisely the poles of its Schwarzian derivative. Finally, we study the pole-dependency of these Schwarzian derivatives. We show that, in the cubic case with simple critical points, an analytic dependency fails precisely when the poles are displaced at the vertices of a regular ideal tetrahedron of the hyperbolic 3-ball. Schwarzian derivatives (dpeaa)DE-He213 rational maps (dpeaa)DE-He213 critical points (dpeaa)DE-He213 meromorphic quadratic differentials (dpeaa)DE-He213 Gao, Yan verfasserin aut Rugh, Hans Henrik verfasserin aut Tan, Lei verfasserin aut Enthalten in Science in China Asheville, NC : Science in China Press, 1995 59(2016), 7 vom: 29. März, Seite 1267-1284 (DE-627)325695059 (DE-600)2038800-7 1862-2763 nnns volume:59 year:2016 number:7 day:29 month:03 pages:1267-1284 https://dx.doi.org/10.1007/s11425-016-5140-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-AST SSG-OPC-ASE GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_152 GBV_ILN_161 GBV_ILN_171 GBV_ILN_187 GBV_ILN_224 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 30.00 ASE 31.00 ASE AR 59 2016 7 29 03 1267-1284
allfieldsSound	10.1007/s11425-016-5140-7 doi (DE-627)SPR019143370 (SPR)s11425-016-5140-7-e DE-627 ger DE-627 rakwb eng 510 530 520 ASE 30.00 bkl 31.00 bkl Cui, GuiZhen verfasserin aut Rational maps as Schwarzian primitives 2016 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract We examine when a meromorphic quadratic differential ϕ with prescribed poles is the Schwarzian derivative of a rational map. We give a necessary and sufficient condition: In the Laurent series of ϕ around each pole c, the most singular term should take the form (1 − d2)/(2(z − c)2), where d is an integer, and then a certain determinant in the next d coefficients should vanish. This condition can be optimized by neglecting some information on one of the poles (i.e., by only requiring it to be a double pole). The case d = 2 was treated by Eremenko (2012). We show that a geometric interpretation of our condition is that the complex projective structure induced by ϕ outside the poles has a trivial holonomy group. This statement was suggested to us by Thurston in a private communication. Our work is related to the problem of finding a rational map f with a prescribed set of critical points, since the critical points of f are precisely the poles of its Schwarzian derivative. Finally, we study the pole-dependency of these Schwarzian derivatives. We show that, in the cubic case with simple critical points, an analytic dependency fails precisely when the poles are displaced at the vertices of a regular ideal tetrahedron of the hyperbolic 3-ball. Schwarzian derivatives (dpeaa)DE-He213 rational maps (dpeaa)DE-He213 critical points (dpeaa)DE-He213 meromorphic quadratic differentials (dpeaa)DE-He213 Gao, Yan verfasserin aut Rugh, Hans Henrik verfasserin aut Tan, Lei verfasserin aut Enthalten in Science in China Asheville, NC : Science in China Press, 1995 59(2016), 7 vom: 29. März, Seite 1267-1284 (DE-627)325695059 (DE-600)2038800-7 1862-2763 nnns volume:59 year:2016 number:7 day:29 month:03 pages:1267-1284 https://dx.doi.org/10.1007/s11425-016-5140-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-AST SSG-OPC-ASE GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_152 GBV_ILN_161 GBV_ILN_171 GBV_ILN_187 GBV_ILN_224 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 30.00 ASE 31.00 ASE AR 59 2016 7 29 03 1267-1284
language	English
source	Enthalten in Science in China 59(2016), 7 vom: 29. März, Seite 1267-1284 volume:59 year:2016 number:7 day:29 month:03 pages:1267-1284
sourceStr	Enthalten in Science in China 59(2016), 7 vom: 29. März, Seite 1267-1284 volume:59 year:2016 number:7 day:29 month:03 pages:1267-1284
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topic_facet	Schwarzian derivatives rational maps critical points meromorphic quadratic differentials
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authorswithroles_txt_mv	Cui, GuiZhen @@aut@@ Gao, Yan @@aut@@ Rugh, Hans Henrik @@aut@@ Tan, Lei @@aut@@
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We give a necessary and sufficient condition: In the Laurent series of ϕ around each pole c, the most singular term should take the form (1 − d2)/(2(z − c)2), where d is an integer, and then a certain determinant in the next d coefficients should vanish. This condition can be optimized by neglecting some information on one of the poles (i.e., by only requiring it to be a double pole). The case d = 2 was treated by Eremenko (2012). We show that a geometric interpretation of our condition is that the complex projective structure induced by ϕ outside the poles has a trivial holonomy group. This statement was suggested to us by Thurston in a private communication. Our work is related to the problem of finding a rational map f with a prescribed set of critical points, since the critical points of f are precisely the poles of its Schwarzian derivative. Finally, we study the pole-dependency of these Schwarzian derivatives. 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author	Cui, GuiZhen
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topic	ddc 510 bkl 30.00 bkl 31.00 misc Schwarzian derivatives misc rational maps misc critical points misc meromorphic quadratic differentials
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title	Rational maps as Schwarzian primitives
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title_full	Rational maps as Schwarzian primitives
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title_sort	rational maps as schwarzian primitives
title_auth	Rational maps as Schwarzian primitives
abstract	Abstract We examine when a meromorphic quadratic differential ϕ with prescribed poles is the Schwarzian derivative of a rational map. We give a necessary and sufficient condition: In the Laurent series of ϕ around each pole c, the most singular term should take the form (1 − d2)/(2(z − c)2), where d is an integer, and then a certain determinant in the next d coefficients should vanish. This condition can be optimized by neglecting some information on one of the poles (i.e., by only requiring it to be a double pole). The case d = 2 was treated by Eremenko (2012). We show that a geometric interpretation of our condition is that the complex projective structure induced by ϕ outside the poles has a trivial holonomy group. This statement was suggested to us by Thurston in a private communication. Our work is related to the problem of finding a rational map f with a prescribed set of critical points, since the critical points of f are precisely the poles of its Schwarzian derivative. Finally, we study the pole-dependency of these Schwarzian derivatives. We show that, in the cubic case with simple critical points, an analytic dependency fails precisely when the poles are displaced at the vertices of a regular ideal tetrahedron of the hyperbolic 3-ball.
abstractGer	Abstract We examine when a meromorphic quadratic differential ϕ with prescribed poles is the Schwarzian derivative of a rational map. We give a necessary and sufficient condition: In the Laurent series of ϕ around each pole c, the most singular term should take the form (1 − d2)/(2(z − c)2), where d is an integer, and then a certain determinant in the next d coefficients should vanish. This condition can be optimized by neglecting some information on one of the poles (i.e., by only requiring it to be a double pole). The case d = 2 was treated by Eremenko (2012). We show that a geometric interpretation of our condition is that the complex projective structure induced by ϕ outside the poles has a trivial holonomy group. This statement was suggested to us by Thurston in a private communication. Our work is related to the problem of finding a rational map f with a prescribed set of critical points, since the critical points of f are precisely the poles of its Schwarzian derivative. Finally, we study the pole-dependency of these Schwarzian derivatives. We show that, in the cubic case with simple critical points, an analytic dependency fails precisely when the poles are displaced at the vertices of a regular ideal tetrahedron of the hyperbolic 3-ball.
abstract_unstemmed	Abstract We examine when a meromorphic quadratic differential ϕ with prescribed poles is the Schwarzian derivative of a rational map. We give a necessary and sufficient condition: In the Laurent series of ϕ around each pole c, the most singular term should take the form (1 − d2)/(2(z − c)2), where d is an integer, and then a certain determinant in the next d coefficients should vanish. This condition can be optimized by neglecting some information on one of the poles (i.e., by only requiring it to be a double pole). The case d = 2 was treated by Eremenko (2012). We show that a geometric interpretation of our condition is that the complex projective structure induced by ϕ outside the poles has a trivial holonomy group. This statement was suggested to us by Thurston in a private communication. Our work is related to the problem of finding a rational map f with a prescribed set of critical points, since the critical points of f are precisely the poles of its Schwarzian derivative. Finally, we study the pole-dependency of these Schwarzian derivatives. We show that, in the cubic case with simple critical points, an analytic dependency fails precisely when the poles are displaced at the vertices of a regular ideal tetrahedron of the hyperbolic 3-ball.
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title_short	Rational maps as Schwarzian primitives
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We give a necessary and sufficient condition: In the Laurent series of ϕ around each pole c, the most singular term should take the form (1 − d2)/(2(z − c)2), where d is an integer, and then a certain determinant in the next d coefficients should vanish. This condition can be optimized by neglecting some information on one of the poles (i.e., by only requiring it to be a double pole). The case d = 2 was treated by Eremenko (2012). We show that a geometric interpretation of our condition is that the complex projective structure induced by ϕ outside the poles has a trivial holonomy group. This statement was suggested to us by Thurston in a private communication. Our work is related to the problem of finding a rational map f with a prescribed set of critical points, since the critical points of f are precisely the poles of its Schwarzian derivative. Finally, we study the pole-dependency of these Schwarzian derivatives. 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Rational maps as Schwarzian primitives

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