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Applications of Quaternionic Holomorphic Geometry to minimal surfaces

K. Leschke, K. Moriya (2016)

Complex Manifolds

In this paper we give a survey of methods of Quaternionic Holomorphic Geometry and of applications of the theory to minimal surfaces. We discuss recent developments in minimal surface theory using integrable systems. In particular, we give the Lopez–Ros deformation and the simple factor dressing in terms of the Gauss map and the Hopf differential of the minimal surface. We illustrate the results for well–known examples of minimal surfaces, namely the Riemann minimal surfaces and the Costa surface....

Applications of the Hadamard product in geometric function theory

Zbigniew Jerzy Jakubowski, Piotr Liczberski, Łucja Żywień (1991)

Mathematica Bohemica

Let 𝒜 denote the set of functions F holomorphic in the unit disc, normalized clasically: F ( 0 ) = 0 , F ' ( 0 ) = 1 , whereas A ⊂ 𝒜 is an arbitrarily fixed subset. In this paper various properties of the classes A α , α ∈ C { - 1 , - 1 2 , ... } , of functions of the form f = F * k α are studied, where F ∈ . A , k α ( z ) = k ( z , α ) = z + 1 1 + α z 2 + ... + 1 1 + ( n - 1 ) α z n + ... , and F * k α denotes the Hadamard product of the functions F and k α . Some special cases of the set A were considered by other authors (see, for example, [15],[6],[3]).

Applications of the Owa-Srivastava Operator to the Class of K-Uniformly Convex Functions

Mishra, A. K., Gochhayat, P. (2006)

Fractional Calculus and Applied Analysis

2000 Mathematics Subject Classification: Primary 30C45, 26A33; Secondary 33C15By making use of the fractional differential operator Ω^λz (0 ≤ λ < 1) due to Owa and Srivastava, a new subclass of univalent functions denoted by k−SPλ (0 ≤ k < ∞) is introduced. The class k−SPλ unifies the concepts of k-uniformly convex functions and k-starlike functions. Certain basic properties of k − SPλ such as inclusion theorem, subordination theorem, growth theorem and class preserving transforms are studied.*...

Applications of the p -adic Nevanlinna theory to functional equations

Abdelbaki Boutabaa, Alain Escassut (2000)

Annales de l'institut Fourier

Let K be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value. We apply the p -adic Nevanlinna theory to functional equations of the form g = R ∘ f , where R ∈ K ( x ) , f , g are meromorphic functions in K , or in an “open disk”, g satisfying conditions on the order of its zeros and poles. In various cases we show that f and g must be constant when they are meromorphic in all K , or they must be quotients of bounded functions when they are meromorphic in an “open disk”. In particular,...

Applications of the theory of differential subordination for functions with fixed initial coefficient to univalent functions

Sumit Nagpal, V. Ravichandran (2012)

Annales Polonici Mathematici

By using the theory of first-order differential subordination for functions with fixed initial coefficient, several well-known results for subclasses of univalent functions are improved by restricting the functions to have fixed second coefficient. The influence of the second coefficient of univalent functions becomes evident in the results obtained.

Approximate roots of pseudo-Anosov diffeomorphisms

T. M. Gendron (2009)

Annales de l’institut Fourier

The Root Conjecture predicts that every pseudo-Anosov diffeomorphism of a closed surface has Teichmüller approximate n th roots for all n ≥ 2 . In this paper, we replace the Teichmüller topology by the heights-widths topology – that is induced by convergence of tangent quadratic differentials with respect to both the heights and widths functionals – and show that every pseudo-Anosov diffeomorphism of a closed surface has heights-widths approximate n th roots for all n ≥ 2 .

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