Rigidity for convex mappings of Reinhardt domains and its applications
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摘要
In this paper, we investigate rigidity and its applications to extreme points of biholomorphic convex mappings on Reinhardt domains. By introducing a version of the scaling method, we precisely construct many unbounded convex mappings with only one in?nite discontinuity on the boundary of this domain. We also give a rigidity of these unbounded convex mappings via the Kobayashi metric and the Liouville-type theorem of entire functions. As an application we obtain a collection of extreme points for the class of normalized convex mappings. Our results extend both the rigidity of convex mappings and related extreme points from the unit ball to Reinhardt domains.
In this paper, we investigate rigidity and its applications to extreme points of biholomorphic convex mappings on Reinhardt domains. By introducing a version of the scaling method, we precisely construct many unbounded convex mappings with only one in?nite discontinuity on the boundary of this domain. We also give a rigidity of these unbounded convex mappings via the Kobayashi metric and the Liouville-type theorem of entire functions. As an application we obtain a collection of extreme points for the class of normalized convex mappings. Our results extend both the rigidity of convex mappings and related extreme points from the unit ball to Reinhardt domains.
In this paper, we investigate rigidity and its applications to extreme points of biholomorphic convex mappings on Reinhardt domains. By introducing a version of the scaling method, we precisely construct many unbounded convex mappings with only one in?nite discontinuity on the boundary of this domain. We also give a rigidity of these unbounded convex mappings via the Kobayashi metric and the Liouville-type theorem of entire functions. As an application we obtain a collection of extreme points for the class of normalized convex mappings. Our results extend both the rigidity of convex mappings and related extreme points from the unit ball to Reinhardt domains.
引文
1 Abate M,Raissy J.Wolff-Denjoy theorems in nonsmooth convex domains.Ann Mat Pura Appl(4),2014,193:1503-1518
2 Bracci F,Gaussier H.A proof of the Muir-Suffridge conjecture for convex maps of the unit ball in Cn.Math Ann,2017,11:1-14
3 Brickman L,MacGregor T,Wilken D.Convex hulls of some classical families of univalent functions.Trans Amer Math Soc,1971,156:91-107
4 Frankel S.Complex geometry of convex domains that cover varieties.Acta Math,1989,163:109-149
5 Graham I,Hamada H,Kohr G,et al.Extremal properties associated with univalent subordination chains in Cn.Math Ann,2014,359:61-99
6 Graham I,Kohr G.Geometric Function Theory in One and Higher Dimensions.New York:Marcel Dekker,2003
7 Gong S.Convex and Starlike Mappings in Several Complex Variables.Beijing-New York:Science Press/Kluwer Academic Publishers,1998
8 Gong S,Liu T.On Roper-Suffridge extension operator.J Anal Math,2002,88:397-404
9 Huang X.On a linearity problem of proper holomorphic mappings between balls in complex spaces of different dimensions.J Differential Geom,1999,51:13-33
10 Huang X.On a semi-rigidity property for holomorphic maps.Asian J Math,2003,7:463-492
11 Huang X,Ji S.Mapping Bn into B2n-1.Invent Math,2001,145:219-250
12 Huang X,Ji S,Yin W.On the third gap for proper holomorphic maps between balls.Math Ann,2014,358:115-142
13 Jarnicki M,Pflug P.Invariant Distances and Metrics in Complex Analysis.Berlin-New York:Walter de Gruyter,1993
14 Kol˙aˇr M.Convexifiability and supporting functions in C2.Math Res Lett,1995,4:505-513
15 Krein M,Milman D.On extreme points of regular convex sets.Studia Math,1940,9:133-138
16 Liu T,Ren G.Decomposition theorem of normalized biholomorphic convex mapping.J Reine Angew Math,1998,496:1-13
17 Liu T,Wang J,Tang X.Schwarz lemma at the boundary of the unit ball in Cnand its applications.J Geom Anal,2015,25:1890-1914
18 Mok N,Tsai I-H.Rigidity of convex realizations of irreducible bounded symmetric domains of rank 2.J Reine Angew Math,1992,431:91-122
19 Muir J R,Suffridge T J.Unbounded convex mappings of the ball in Cn.Proc Amer Math Soc,2001,129:3389-3393
20 Muir J R,Suffridge T J.Extreme points for convex mappings of Bn.J Anal Math,2006,98:169-182
21 Muir J R,Suffridge T J.A generlization of the half-plane mappings to the ball in Cn.Trans Amer Math Soc,2007,359:1485-1498
22 Pinchuk S.The scaling method and holomorphic mappings.In:Several Complex Variables and Complex Geometry,Part 1.Proceedings of Symposia in Pure Mathematics,vol.52.Providence:Amer Math Soc,1991,151-161
23 Suffridge T J.The principle of subordination applied to functions of several variables.Pacific J Math,1970,1:241-248
24 Sunada T.Holomorphic equivalence problem for bounded Reinhardt domains.Math Ann,1978,235:111-128
25 Tang X,Liu T.The Schwarz lemma at the boundary of the egg domain Bp1,p2in Cn.Canad Math Bull,2015,58:381-392
26 Thullen P.Zur den Abbildungen durch analytische Funktionen mehrerer komplexer Ver ander-lichen.Math Ann,1931,104:244-259
27 Wang J,Liu T.The Roper-Suffridge extension operator and its applications to convex mappings in C2.Trans Amer Math Soc,2018,370:7743-7759
28 Zhang W,Liu T.On decomposition theorem of normalized biholomorphic convex mappings in Reinhardt domains.Sci China Ser A,2003,46:94-106
2 Bracci F,Gaussier H.A proof of the Muir-Suffridge conjecture for convex maps of the unit ball in Cn.Math Ann,2017,11:1-14
3 Brickman L,MacGregor T,Wilken D.Convex hulls of some classical families of univalent functions.Trans Amer Math Soc,1971,156:91-107
4 Frankel S.Complex geometry of convex domains that cover varieties.Acta Math,1989,163:109-149
5 Graham I,Hamada H,Kohr G,et al.Extremal properties associated with univalent subordination chains in Cn.Math Ann,2014,359:61-99
6 Graham I,Kohr G.Geometric Function Theory in One and Higher Dimensions.New York:Marcel Dekker,2003
7 Gong S.Convex and Starlike Mappings in Several Complex Variables.Beijing-New York:Science Press/Kluwer Academic Publishers,1998
8 Gong S,Liu T.On Roper-Suffridge extension operator.J Anal Math,2002,88:397-404
9 Huang X.On a linearity problem of proper holomorphic mappings between balls in complex spaces of different dimensions.J Differential Geom,1999,51:13-33
10 Huang X.On a semi-rigidity property for holomorphic maps.Asian J Math,2003,7:463-492
11 Huang X,Ji S.Mapping Bn into B2n-1.Invent Math,2001,145:219-250
12 Huang X,Ji S,Yin W.On the third gap for proper holomorphic maps between balls.Math Ann,2014,358:115-142
13 Jarnicki M,Pflug P.Invariant Distances and Metrics in Complex Analysis.Berlin-New York:Walter de Gruyter,1993
14 Kol˙aˇr M.Convexifiability and supporting functions in C2.Math Res Lett,1995,4:505-513
15 Krein M,Milman D.On extreme points of regular convex sets.Studia Math,1940,9:133-138
16 Liu T,Ren G.Decomposition theorem of normalized biholomorphic convex mapping.J Reine Angew Math,1998,496:1-13
17 Liu T,Wang J,Tang X.Schwarz lemma at the boundary of the unit ball in Cnand its applications.J Geom Anal,2015,25:1890-1914
18 Mok N,Tsai I-H.Rigidity of convex realizations of irreducible bounded symmetric domains of rank 2.J Reine Angew Math,1992,431:91-122
19 Muir J R,Suffridge T J.Unbounded convex mappings of the ball in Cn.Proc Amer Math Soc,2001,129:3389-3393
20 Muir J R,Suffridge T J.Extreme points for convex mappings of Bn.J Anal Math,2006,98:169-182
21 Muir J R,Suffridge T J.A generlization of the half-plane mappings to the ball in Cn.Trans Amer Math Soc,2007,359:1485-1498
22 Pinchuk S.The scaling method and holomorphic mappings.In:Several Complex Variables and Complex Geometry,Part 1.Proceedings of Symposia in Pure Mathematics,vol.52.Providence:Amer Math Soc,1991,151-161
23 Suffridge T J.The principle of subordination applied to functions of several variables.Pacific J Math,1970,1:241-248
24 Sunada T.Holomorphic equivalence problem for bounded Reinhardt domains.Math Ann,1978,235:111-128
25 Tang X,Liu T.The Schwarz lemma at the boundary of the egg domain Bp1,p2in Cn.Canad Math Bull,2015,58:381-392
26 Thullen P.Zur den Abbildungen durch analytische Funktionen mehrerer komplexer Ver ander-lichen.Math Ann,1931,104:244-259
27 Wang J,Liu T.The Roper-Suffridge extension operator and its applications to convex mappings in C2.Trans Amer Math Soc,2018,370:7743-7759
28 Zhang W,Liu T.On decomposition theorem of normalized biholomorphic convex mappings in Reinhardt domains.Sci China Ser A,2003,46:94-106