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Schwarz's Lemma
In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex differential geometry that estimates the (squared) pointwise norm , \partial f , ^2 of a holomorphic map f:(X,g_X) \to (Y,g_Y) between Hermitian manifolds under curvature assumptions on g_X and g_Y. The classical Schwarz lemma is a result in complex analysis typically viewed to be about holomorphic functions from the open set, open unit disk \mathbb := \to itself. The Schwarz lemma has opened several branches of complex geometry, and become an essential tool in the use of geometric PDE methods in complex geometry. Statement of the classical Schwarz Lemma Let \mathbf = \ be the open unit disk in the complex number, complex plane \mathbb centered at the origin (mathematics), origin, and let f : \mathbf\rightarrow \mathbb be a holomorphic map such that f(0) = 0 and , f(z), \leq 1 on \mathbf. Then , f(z), \leq , z, for all z \in \mathbf, and , f'(0), \leq 1. Moreover, if , f(z), = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Möbius Transformation
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z) = \frac of one complex number, complex variable ; here the coefficients , , , are complex numbers satisfying . Geometrically, a Möbius transformation can be obtained by first applying the inverse stereographic projection from the plane to the unit sphere, moving and rotating the sphere to a new location and orientation in space, and then applying a stereographic projection to map from the sphere back to the plane. These transformations preserve angles, map every straight line to a line or circle, and map every circle to a line or circle. The Möbius transformations are the projective transformations of the complex projective line. They form a group (mathematics), group called the Möbius group, which is the projective linear group . Together with its subgroups, it has numerous applications in mathematics and physics. Möbius geometry, Möbius geometries and t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Surfaces
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together. Examples of Riemann surfaces include graphs of multivalued functions such as √''z'' or log(''z''), e.g. the subset of pairs with . Every Riemann surface is a surface: a two-dimensional real manifold, but it contains more structure (specifically a complex structure). Conversely, a two-dimensional real manifold can be turned into a Riemann surface (usually in several inequivalent ways) if and only if it is orientable and metrizable. Given this, the sphere and torus admit complex structures but the Möbius ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bloch's Principle
Bloch's principle is a philosophical principle in mathematics stated by André Bloch. Bloch states the principle in Latin as: ''Nihil est in infinito quod non prius fuerit in finito,'' and explains this as follows: Every proposition in whose statement the actual infinity occurs can be always considered a consequence, almost immediate, of a proposition where it does not occur, a proposition in ''finite terms''. Bloch mainly applied this principle to the theory of functions of a complex variable. Thus, for example, according to this principle, Picard's theorem corresponds to Schottky's theorem, and Valiron's theorem corresponds to Bloch's theorem. Based on his Principle, Bloch was able to predict or conjecture several important results such as the Ahlfors's Five Islands theorem, Cartan's theorem on holomorphic curves omitting hyperplanes, Hayman's result that an exceptional set of radii is unavoidable in Nevanlinna theory. In the more recent times several general theorems w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kobayashi Hyperbolic
In mathematics and especially complex geometry, the Kobayashi metric is a pseudometric intrinsically associated to any complex manifold. It was introduced by Shoshichi Kobayashi in 1967. Kobayashi hyperbolic manifolds are an important class of complex manifolds, defined by the property that the Kobayashi pseudometric is a metric. Kobayashi hyperbolicity of a complex manifold ''X'' implies that every holomorphic map from the complex line C to ''X'' is constant. Definition The origins of the concept lie in Schwarz's lemma in complex analysis. Namely, if ''f'' is a holomorphic function on the open unit disc ''D'' in the complex numbers C such that ''f''(0) = 0 and , ''f''(''z''), 0 for all ''x'' ≠ ''y'' in ''X''. Informally, this means that there is a genuine bound on the size of discs mapping holomorphically into ''X''. In these terms, Schwarz's lemma says that the unit disc ''D'' is Kobayashi hyperbolic, and more precisely that the Kobayashi metric on ''D'' is exactly the Poinc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nevanlinna–Pick Interpolation
In complex analysis, given ''initial data'' consisting of n points \lambda_1, \ldots, \lambda_n in the complex unit disk \mathbb and ''target data'' consisting of n points z_1, \ldots, z_n in \mathbb, the Nevanlinna–Pick interpolation problem is to find a holomorphic function \varphi that interpolates the data, that is for all i \in \, :\varphi(\lambda_i) = z_i, subject to the constraint \left\vert \varphi(\lambda) \right\vert \le 1 for all \lambda \in \mathbb. Georg Pick and Rolf Nevanlinna solved the problem independently in 1916 and 1919 respectively, showing that an interpolating function exists if and only if a matrix defined in terms of the initial and target data is positive semi-definite. Background The Nevanlinna–Pick theorem represents an n-point generalization of the Schwarz lemma. The invariant form of the Schwarz lemma states that for a holomorphic function f:\mathbb\to\mathbb, for all \lambda_1, \lambda_2 \in \mathbb, : \left, \frac\ \leq \left, \frac\. Settin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Koebe 1/4 Theorem
In complex analysis, a branch of mathematics, the Koebe 1/4 theorem states the following: Koebe Quarter Theorem. The image of an injective analytic function f:\mathbf\to\mathbb from the unit disk \mathbf onto a subset of the complex plane contains the disk whose center is f(0) and whose radius is , f'(0), /4. The theorem is named after Paul Koebe, who conjectured the result in 1907. The theorem was proven by Ludwig Bieberbach in 1916. The example of the Koebe function shows that the constant 1/4 in the theorem cannot be improved (increased). A related result is the Schwarz lemma, and a notion related to both is conformal radius. Grönwall's area theorem Suppose that :g(z) = z +b_1z^ + b_2 z^ + \cdots is univalent in , z, >1. Then :\sum_^\infty n, b_n, ^2 \le 1. In fact, if r > 1, the complement of the image of the disk , z, >r is a bounded domain X(r). Its area is given by : \int_ dx\,dy = \int_\overline\,dz = \int_\overline\,dg=\pi r^2 - \pi\sum_^\infty n, b_n, ^2 r^. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Univalent Mapping
In mathematics, in the branch of complex analysis, a holomorphic function on an open subset of the complex plane is called univalent if it is injective. Examples The function f \colon z \mapsto 2z + z^2 is univalent in the open unit disc, as f(z) = f(w) implies that f(z) - f(w) = (z-w)(z+w+2) = 0. As the second factor is non-zero in the open unit disc, z = w so f is injective. Basic properties One can prove that if G and \Omega are two open connected sets in the complex plane, and :f: G \to \Omega is a univalent function such that f(G) = \Omega (that is, f is surjective), then the derivative of f is never zero, f is invertible, and its inverse f^ is also holomorphic. More, one has by the chain rule :(f^)'(f(z)) = \frac for all z in G. Comparison with real functions For real analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Injective
In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, implies ). In other words, every element of the function's codomain is the image of one element of its domain. The term must not be confused with that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain. A homomorphism between algebraic structures is a function that is compatible with the operations of the structures. For all common algebraic structures, and, in particular for vector spaces, an is also called a . However, in the more general context of category theory, the definition of a monomorphism differs from that of an injective homomorphism. This is thus a theorem that they are equivalent for algebraic structures; see for more details. A func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Branges' Theorem
In complex analysis, de Branges's theorem, or the Bieberbach conjecture, is a theorem that gives a necessary condition on a holomorphic function in order for it to map the open unit disk of the complex plane injectively to the complex plane. It was posed by and finally proven by . The statement concerns the Taylor coefficients a_n of a univalent function, i.e. a one-to-one holomorphic function that maps the unit disk into the complex plane, normalized as is always possible so that a_0=0 and a_1=1. That is, we consider a function defined on the open unit disk which is holomorphic and injective ('' univalent'') with Taylor series of the form :f(z)=z+\sum_ a_n z^n. Such functions are called ''schlicht''. The theorem then states that : , a_n, \leq n \quad \textn\geq 2. The Koebe function (see below) is a function for which a_n=n for all n, and it is schlicht, so we cannot find a stricter limit on the absolute value of the nth coefficient. Schlicht functions The normalization ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schwarz–Ahlfors–Pick Theorem
In mathematics, the Schwarz–Ahlfors–Pick theorem is an extension of the Schwarz lemma for hyperbolic geometry, such as the Poincaré half-plane model. The Schwarz–Pick lemma states that every holomorphic function from the unit disk ''U'' to itself, or from the upper half-plane ''H'' to itself, will not increase the Poincaré distance between points. The unit disk ''U'' with the Poincaré metric has negative Gaussian curvature −1. In 1938, Lars Ahlfors generalised the lemma to maps from the unit disk to other negatively curved surfaces: Theorem ( Schwarz– Ahlfors– Pick). Let ''U'' be the unit disk with Poincaré metric \rho; let ''S'' be a Riemann surface endowed with a Hermitian metric \sigma whose Gaussian curvature is ≤ −1; let f:U\rightarrow S be a holomorphic function. Then :\sigma(f(z_1),f(z_2)) \leq \rho(z_1,z_2) for all z_1,z_2 \in U. A generalization of this theorem was proved by Shing-Tung Yau Shing-Tung Yau (; ; born April 4, 1949) i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Difference Quotient
In single-variable calculus, the difference quotient is usually the name for the expression : \frac which when taken to the Limit of a function, limit as ''h'' approaches 0 gives the derivative of the Function (mathematics), function ''f''. The name of the expression stems from the fact that it is the quotient of the Difference (mathematics), difference of values of the function by the difference of the corresponding values of its argument (the latter is (''x'' + ''h'') - ''x'' = ''h'' in this case). The difference quotient is a measure of the average rate of change (mathematics), rate of change of the function over an Interval (mathematics), interval (in this case, an interval of length ''h''). The limit of the difference quotient (i.e., the derivative) is thus the instantaneous rate of change. By a slight change in notation (and viewpoint), for an interval [''a'', ''b''], the difference quotient : \frac is called the mean (or average) value of the derivative of ''f'' over th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
