We study the Carathéodory metric on some generalized Teichmüller spaces. Our paper is especially ... more We study the Carathéodory metric on some generalized Teichmüller spaces. Our paper is especially inspired by the papers by Earle (1974) and Miyachi (2006). Earle (1974) showed that the Carathéodory metric is complete on any Teichmüller space. Miyachi (2006) extended this result for asymptotic Teichmüller spaces. We study the completeness of the Carathéodory metric on product Teichmüller spaces and on the Teichmüller space of a closed set in the Riemann sphere. Eräiden yleistettyjen Teichmüllerin avaruuksien Carathéodoryn metriikka Tiivistelmä. Tutkimme eräiden yleistettyjen Teichmüllerin avaruuksien Carathéodoryn metriikkaa. Tarkasteluamme inspiroivat erityisesti Earlen (1974) ja Miyachin (2006) työt. Earle (1974) osoitti, että jokaisen Teichmüllerin avaruuden Carathéodoryn metriikka on täydellinen. Miyachi (2006) yleisti tämän tuloksen asymptoottisiin Teichmüllerin avaruuksiin. Me tutkimme Teichmüllerin tuloavaruuksien ja Riemannin pallon suljetun joukon Teichmüllerin tuloavaruuden Carathéodoryn metriikan täydellisyyttä.
We discuss the Carathéodory and Kobayashi pseudometrics and their infinitesimal forms on complex ... more We discuss the Carathéodory and Kobayashi pseudometrics and their infinitesimal forms on complex Banach manifolds. Our discussiion includes a very elementary treatment of the Kobayashi pseudometric as an integral of its infinitesimal form. We also prove new distortion theorems for the Carathéodory pseudometric under holomorphic maps from the open unit disk to a complex Banach manifold.
Techmüller theory and moduli problems : proceedings of the U.S.-India workshop in Techmüller theory and moduli problems, Harish-Chandra Research Institute, Allahabad, January 2006
Douady-Early section, holomorphic motions, and some applications
We review several applications of Douady-Earle sections to holomorphic motions over infinite dime... more We review several applications of Douady-Earle sections to holomorphic motions over infinite dimensional parameter spaces. Using Douady-Earle sections we study group-equivariant extensions of holomorphic motions. We also discuss the relationship between extending holomorphic motions and lifting holomorphic maps. Finally, we discuss several applications of holomorphic motions in Complex analysis.
Extensions of holomorphic motions of quasiconformal motions
We reinterpret our results in [Isr. J. Math. 159, 277–288 (2007; Zbl 1119.30024)] by using quasic... more We reinterpret our results in [Isr. J. Math. 159, 277–288 (2007; Zbl 1119.30024)] by using quasiconformal motions of the sphere, in the sense of Sullivan and Thurston. We show that a normalized holomorphic motion of a closed set E in the Riemann sphere extends to a normalized quasiconformal motion of the sphere if and only if the holomorphic motion is induced by a holomorphic map into the Teichmüller space of E, denoted by T(E). We also study a “universal” quasiconformal motion of the sphere and some other properties of continuous and quasiconformal motions.
We review several applications of Douady-Earle section to holomorphic motions over infinite dimen... more We review several applications of Douady-Earle section to holomorphic motions over infinite dimensional parameter spaces. Using Douady-Earle section we study group-equivariant extensions of holomorphic motions. We also discuss the relationship between extending holomorphic motions and lifting holomorphic maps. Finally, we discuss several applications of holomorphic motions in complex analysis.
We use Earle's generalization of Montel's theorem to obtain some results on holomorphic motions o... more We use Earle's generalization of Montel's theorem to obtain some results on holomorphic motions over infinite dimensional parameter spaces. We also study some properties of group-equivariant extensions of holomorphic motions.
We give an easy description of the barycentric extension of a map of the unit circle to the close... more We give an easy description of the barycentric extension of a map of the unit circle to the closed unit disk using some ideas from dynamical systems. We then prove that every circle endomorphism of the unit circle of degree d b 2 (with a topological expansion condition) has a conformally natural extension to the closed unit disk which is real analytic on the open unit disk. If the endomorphism is uniformly quasisymmetric, then the extension is quasiconformal. 1. Conformal barycenter Let C denote the complex plane, T ¼ fz A C : jzj ¼ 1g the unit circle, D ¼ fz A C : jzj < 1g the open unit disk, and D ¼ D U T the closed unit disk. Let m be a probability measure on T. The barycenter of m, by definition, is BCðmÞ ¼ ð T x dmðxÞ: Suppose f : T ! T is a m-measurable map. The pushforward measure f à m of m by f , by definition, is f à mðEÞ ¼ mð f À1 ðEÞÞ for any m-measurable set E on T. For two m-measurable maps f and g from T to T, the composition f g is also a m-measurable map from T to T. Then ð f gÞ Ã m ¼ f à g à m since f à g à mðEÞ ¼ g à mð f À1 ðEÞÞ ¼ mðg À1 ð f À1 ðEÞÞÞ ¼ mðð f gÞ À1 ðEÞÞ ¼ ð f gÞ Ã mðEÞ for any m-measurable set E J T.
The subject of holomorphic motions over the open unit disc has found important applications in co... more The subject of holomorphic motions over the open unit disc has found important applications in complex dynamics. In this paper, we study holomorphic motions over more general parameter spaces. The Teichmiiller space of a closed subset of the Riemann sphere is shown to be a universal parameter space for holomorphic motions of the set over a simply connected complex Banach manifold. As a consequence, we prove a generalization of the "Harmonic A-Lemma" of Bers and Royden. We also study some other applications.
We generalize the principle of Teichmiiller contraction and deduce the Hamilton-Krushkal conditio... more We generalize the principle of Teichmiiller contraction and deduce the Hamilton-Krushkal condition for extremal quasiconformal mappings in the Teichmiiller space of a closed set in the Riemann sphere.
We prove that a normalized holomorphic motion of a closed set E is induced by a holomorphic map i... more We prove that a normalized holomorphic motion of a closed set E is induced by a holomorphic map into the Teichmüller space of E, denoted by T (E), if and only if it can be extended to a normalized continuous motion of the Riemann sphere. We also prove that the extension can be chosen to have additional properties.
In their paper [17], Sullivan and Thurston introduced the notion of quasiconformal motions, and p... more In their paper [17], Sullivan and Thurston introduced the notion of quasiconformal motions, and proved an extension theorem for quasiconformal motions over an interval. We prove some new properties of (normalized) quasiconformal motions of a closed set E in the Riemann sphere, over connected Hausdorff spaces. As a spin-off, we strengthen the result of Sullivan and Thurston, and show that if a quasiconformal motion of E over an interval has a certain group-equivariance property, then the extended quasiconformal motion can be chosen to have the same group-equivariance property. Our main theorem proves a result on isomorphisms of continuous families of Möbius groups arising from a group-equivariant quasiconformal motion of E over a path-connected Hausdorff space. Our techniques connect the Teichmüller space of the closed set E with quasiconformal motions of E.
For a closed subset E of the Riemann sphere, its Teichmüller space TðEÞ is a universal parameter ... more For a closed subset E of the Riemann sphere, its Teichmüller space TðEÞ is a universal parameter space for holomorphic motions of E over a simply connected complex Banach manifold. In this paper, we study some new applications of this universal property. f à ðfÞðx; zÞ ¼ fð f ðxÞ; zÞ Eðx; zÞ A W  E ð1:1Þ of E over W. Unless otherwise stated, we will assume that E is a closed subset ofĈ C and that 0; 1; y A E. Associated to each such set E inĈ C, there is a contractible complex Banach manifold which we call the Teichmü ller space of the closed set E, denoted by TðEÞ. This was first studied by G. Lieb in his doctoral dissertation (see [15]). We can also define a holomorphic motion 85
We study the Carathéodory metric on some generalized Teichmüller spaces. Our paper is especially ... more We study the Carathéodory metric on some generalized Teichmüller spaces. Our paper is especially inspired by the papers by Earle (1974) and Miyachi (2006). Earle (1974) showed that the Carathéodory metric is complete on any Teichmüller space. Miyachi (2006) extended this result for asymptotic Teichmüller spaces. We study the completeness of the Carathéodory metric on product Teichmüller spaces and on the Teichmüller space of a closed set in the Riemann sphere. Eräiden yleistettyjen Teichmüllerin avaruuksien Carathéodoryn metriikka Tiivistelmä. Tutkimme eräiden yleistettyjen Teichmüllerin avaruuksien Carathéodoryn metriikkaa. Tarkasteluamme inspiroivat erityisesti Earlen (1974) ja Miyachin (2006) työt. Earle (1974) osoitti, että jokaisen Teichmüllerin avaruuden Carathéodoryn metriikka on täydellinen. Miyachi (2006) yleisti tämän tuloksen asymptoottisiin Teichmüllerin avaruuksiin. Me tutkimme Teichmüllerin tuloavaruuksien ja Riemannin pallon suljetun joukon Teichmüllerin tuloavaruuden Carathéodoryn metriikan täydellisyyttä.
We discuss the Carathéodory and Kobayashi pseudometrics and their infinitesimal forms on complex ... more We discuss the Carathéodory and Kobayashi pseudometrics and their infinitesimal forms on complex Banach manifolds. Our discussiion includes a very elementary treatment of the Kobayashi pseudometric as an integral of its infinitesimal form. We also prove new distortion theorems for the Carathéodory pseudometric under holomorphic maps from the open unit disk to a complex Banach manifold.
Techmüller theory and moduli problems : proceedings of the U.S.-India workshop in Techmüller theory and moduli problems, Harish-Chandra Research Institute, Allahabad, January 2006
Douady-Early section, holomorphic motions, and some applications
We review several applications of Douady-Earle sections to holomorphic motions over infinite dime... more We review several applications of Douady-Earle sections to holomorphic motions over infinite dimensional parameter spaces. Using Douady-Earle sections we study group-equivariant extensions of holomorphic motions. We also discuss the relationship between extending holomorphic motions and lifting holomorphic maps. Finally, we discuss several applications of holomorphic motions in Complex analysis.
Extensions of holomorphic motions of quasiconformal motions
We reinterpret our results in [Isr. J. Math. 159, 277–288 (2007; Zbl 1119.30024)] by using quasic... more We reinterpret our results in [Isr. J. Math. 159, 277–288 (2007; Zbl 1119.30024)] by using quasiconformal motions of the sphere, in the sense of Sullivan and Thurston. We show that a normalized holomorphic motion of a closed set E in the Riemann sphere extends to a normalized quasiconformal motion of the sphere if and only if the holomorphic motion is induced by a holomorphic map into the Teichmüller space of E, denoted by T(E). We also study a “universal” quasiconformal motion of the sphere and some other properties of continuous and quasiconformal motions.
We review several applications of Douady-Earle section to holomorphic motions over infinite dimen... more We review several applications of Douady-Earle section to holomorphic motions over infinite dimensional parameter spaces. Using Douady-Earle section we study group-equivariant extensions of holomorphic motions. We also discuss the relationship between extending holomorphic motions and lifting holomorphic maps. Finally, we discuss several applications of holomorphic motions in complex analysis.
We use Earle's generalization of Montel's theorem to obtain some results on holomorphic motions o... more We use Earle's generalization of Montel's theorem to obtain some results on holomorphic motions over infinite dimensional parameter spaces. We also study some properties of group-equivariant extensions of holomorphic motions.
We give an easy description of the barycentric extension of a map of the unit circle to the close... more We give an easy description of the barycentric extension of a map of the unit circle to the closed unit disk using some ideas from dynamical systems. We then prove that every circle endomorphism of the unit circle of degree d b 2 (with a topological expansion condition) has a conformally natural extension to the closed unit disk which is real analytic on the open unit disk. If the endomorphism is uniformly quasisymmetric, then the extension is quasiconformal. 1. Conformal barycenter Let C denote the complex plane, T ¼ fz A C : jzj ¼ 1g the unit circle, D ¼ fz A C : jzj < 1g the open unit disk, and D ¼ D U T the closed unit disk. Let m be a probability measure on T. The barycenter of m, by definition, is BCðmÞ ¼ ð T x dmðxÞ: Suppose f : T ! T is a m-measurable map. The pushforward measure f à m of m by f , by definition, is f à mðEÞ ¼ mð f À1 ðEÞÞ for any m-measurable set E on T. For two m-measurable maps f and g from T to T, the composition f g is also a m-measurable map from T to T. Then ð f gÞ Ã m ¼ f à g à m since f à g à mðEÞ ¼ g à mð f À1 ðEÞÞ ¼ mðg À1 ð f À1 ðEÞÞÞ ¼ mðð f gÞ À1 ðEÞÞ ¼ ð f gÞ Ã mðEÞ for any m-measurable set E J T.
The subject of holomorphic motions over the open unit disc has found important applications in co... more The subject of holomorphic motions over the open unit disc has found important applications in complex dynamics. In this paper, we study holomorphic motions over more general parameter spaces. The Teichmiiller space of a closed subset of the Riemann sphere is shown to be a universal parameter space for holomorphic motions of the set over a simply connected complex Banach manifold. As a consequence, we prove a generalization of the "Harmonic A-Lemma" of Bers and Royden. We also study some other applications.
We generalize the principle of Teichmiiller contraction and deduce the Hamilton-Krushkal conditio... more We generalize the principle of Teichmiiller contraction and deduce the Hamilton-Krushkal condition for extremal quasiconformal mappings in the Teichmiiller space of a closed set in the Riemann sphere.
We prove that a normalized holomorphic motion of a closed set E is induced by a holomorphic map i... more We prove that a normalized holomorphic motion of a closed set E is induced by a holomorphic map into the Teichmüller space of E, denoted by T (E), if and only if it can be extended to a normalized continuous motion of the Riemann sphere. We also prove that the extension can be chosen to have additional properties.
In their paper [17], Sullivan and Thurston introduced the notion of quasiconformal motions, and p... more In their paper [17], Sullivan and Thurston introduced the notion of quasiconformal motions, and proved an extension theorem for quasiconformal motions over an interval. We prove some new properties of (normalized) quasiconformal motions of a closed set E in the Riemann sphere, over connected Hausdorff spaces. As a spin-off, we strengthen the result of Sullivan and Thurston, and show that if a quasiconformal motion of E over an interval has a certain group-equivariance property, then the extended quasiconformal motion can be chosen to have the same group-equivariance property. Our main theorem proves a result on isomorphisms of continuous families of Möbius groups arising from a group-equivariant quasiconformal motion of E over a path-connected Hausdorff space. Our techniques connect the Teichmüller space of the closed set E with quasiconformal motions of E.
For a closed subset E of the Riemann sphere, its Teichmüller space TðEÞ is a universal parameter ... more For a closed subset E of the Riemann sphere, its Teichmüller space TðEÞ is a universal parameter space for holomorphic motions of E over a simply connected complex Banach manifold. In this paper, we study some new applications of this universal property. f à ðfÞðx; zÞ ¼ fð f ðxÞ; zÞ Eðx; zÞ A W  E ð1:1Þ of E over W. Unless otherwise stated, we will assume that E is a closed subset ofĈ C and that 0; 1; y A E. Associated to each such set E inĈ C, there is a contractible complex Banach manifold which we call the Teichmü ller space of the closed set E, denoted by TðEÞ. This was first studied by G. Lieb in his doctoral dissertation (see [15]). We can also define a holomorphic motion 85
Uploads
Papers by Sudeb Mitra