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Simply connected Riemann surface is equivalent to an open disk, complex plane, or sphere
In mathematics, the uniformization theorem states that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces:
Uniformization_theorem
Mathematical theorem
onto D {\displaystyle D} . Koebe's uniformization theorem for normal families also generalizes to yield uniformizers f {\displaystyle f} for multiply-connected
Riemann_mapping_theorem
studied by Koebe who proved in 1910, as a generalization of the uniformization theorem, that every such surface is conformally equivalent to either the
Planar_Riemann_surface
Branch of topology
according to their universal cover. The uniformization theorem is a generalization of the Riemann mapping theorem from proper simply connected open subsets
Low-dimensional_topology
mathematics, the simultaneous uniformization theorem, proved by Bers (1960), states that it is possible to simultaneously uniformize two different Riemann surfaces
Simultaneous uniformization theorem
Simultaneous_uniformization_theorem
Topics referred to by the same term
Look up uniformization in Wiktionary, the free dictionary. Uniformization may refer to: Uniformization (set theory), a mathematical concept in set theory
Uniformization
Mathematical theorem in real analysis
In mathematics, the uniform limit theorem states that the uniform limit of any sequence of continuous functions is continuous. More precisely, let X be
Uniform_limit_theorem
Theorem stating that pointwise boundedness implies uniform boundedness
In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with
Uniform_boundedness_principle
In descriptive set theory the Jankov–von Neumann uniformization theorem is a result saying that every measurable relation on a pair of standard Borel spaces
Jankov–von Neumann uniformization theorem
Jankov–von_Neumann_uniformization_theorem
Mathematics of smooth surfaces
such as the Gauss–Bonnet theorem, the uniformization theorem, the von Mangoldt-Hadamard theorem, and the embeddability theorem. There are other important
Differential geometry of surfaces
Differential_geometry_of_surfaces
Branch of mathematics
Low-dimensional topology is strongly geometric, as reflected in the uniformization theorem in 2 dimensions – every surface admits a constant curvature metric;
Topology
Two-dimensional manifold
geometric proof, which yields a stronger geometric result, is the uniformization theorem. This was originally proven only for Riemann surfaces in the 1880s
Surface_(topology)
Mode of convergence of a function sequence
infamously known as "Cauchy's wrong theorem". The uniform limit theorem shows that a stronger form of convergence, uniform convergence, is needed to ensure
Uniform_convergence
German mathematician (1849–1925)
prove a grand uniformization theorem that would establish the new theory more completely. Klein succeeded in formulating such a theorem and in describing
Felix_Klein
On when a family of real, continuous functions has a uniformly convergent subsequence
bounded interval has a uniformly convergent subsequence. The main condition is the equicontinuity of the family of functions. The theorem is the basis of many
Arzelà–Ascoli_theorem
Partial differential equation
Various uniformization theorems can be proved using the equation, including the measurable Riemann mapping theorem and the simultaneous uniformization theorem
Beltrami_equation
Theorem in algebraic geometry
In the context of compact Riemann surfaces X, via the Riemann uniformization theorem, this can be seen as a distinction between the surfaces of different
Hurwitz's automorphisms theorem
Hurwitz's_automorphisms_theorem
Three dimensional analogue of uniformization conjecture
structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected
Geometrization_conjecture
Branch of mathematics studying (smooth) functions of manifolds
Low-dimensional topology is strongly geometric, as reflected in the uniformization theorem in 2 dimensions – every surface admits a constant curvature metric;
Geometric_topology
On tangency patterns of circles
a circle packing for any planar graph is based on his conformal uniformization theorem, saying that a finitely connected planar domain is conformally equivalent
Circle_packing_theorem
Russian mathematician (born 1966)
Spherical space form conjecture Thurston elliptization conjecture Uniformization theorem The New Yorker authors explained Perelman's reference to "some ugly
Grigori_Perelman
Space where every point locally resembles a hyperbolic space
homeomorphism. This is a consequence of the uniformization theorem for surfaces and the geometrization theorem for 3-manifolds proved by Perelman. A hyperbolic
Hyperbolic_manifold
Mathematical theorem in the study of analysis
Weierstrass approximation theorem states that every continuous function defined on a closed interval [a, b] can be uniformly approximated as closely as
Stone–Weierstrass_theorem
Basic question in geometry and topology
2-dimensional manifold (surface) admits a constant curvature metric, by the uniformization theorem. There are 3 such curvatures (positive, zero, and negative). This
Classification_of_manifolds
Model of the extended complex plane plus a point at infinity
identified with C {\displaystyle \mathbf {C} } . On the other hand, the uniformization theorem, a central result in the classification of Riemann surfaces, states
Riemann_sphere
Theory of probability
theory of probability, the Glivenko–Cantelli theorem (sometimes referred to as the fundamental theorem of statistics), named after Valery Ivanovich Glivenko
Glivenko–Cantelli_theorem
Conformal structure admits a Hodge dual of 1-forms without even specifying a metric
Sobolev spaces. These techniques were originally applied to prove the uniformization theorem and its generalization to planar Riemann surfaces. Later they supplied
Differential forms on a Riemann surface
Differential_forms_on_a_Riemann_surface
theorem Toponogov theorem Sphere theorem Hodge theory Uniformization theorem Yamabe problem Killing vector field Myers-Steenrod theorem Hodge star operator
List of differential geometry topics
List_of_differential_geometry_topics
Study of complex manifolds and several complex variables
Riemann surface. The classification essentially follows from the uniformization theorem, and is as follows: g = 0: C P 1 {\displaystyle \mathbb {CP} ^{1}}
Complex_geometry
Sokhotski–Plemelj theorem (complex analysis) Uniformization theorem (complex analysis, differential geometry) Van Vleck's theorem (mathematical analysis)
List_of_theorems
Concept in algebraic geometry
Zariski used a more roundabout method: he first proved a local uniformization theorem showing that every valuation of a surface could be resolved, then
Resolution_of_singularities
German mathematician (1882–1945)
primarily with complex analysis, his best known results being on the uniformization of Riemann surfaces. Paul Koebe was born 15 February 1882 in Luckenwalde
Paul_Koebe
Fundamental theorem in probability theory and statistics
In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the distribution of a normalized version of the sample
Central_limit_theorem
Sufficiency theorem for reconstructing signals from samples
The Nyquist–Shannon sampling theorem is a theorem in the field of signal processing which serves as a fundamental bridge between continuous-time signals
Nyquist–Shannon sampling theorem
Nyquist–Shannon_sampling_theorem
Two geometries based on axioms closely related to those specifying Euclidean geometry
referred to as parabolic in the context of conformal geometry: see Uniformization theorem. for instance, and Yaglom 1968 a 21st axiom appeared in the French
Non-Euclidean_geometry
whose genus is at least 1 {\displaystyle 1} . The Uniformization theorem and the Gauss–Bonnet theorem can both be applied to orientable Riemann surfaces
Non-positive_curvature
the Poincaré conjecture, and Richard S. Hamilton's proof of the uniformization theorem Calabi flow, a flow for Kähler metrics Yamabe flow Important classes
Geometric_flow
of minimal volume is realized by the metrics appearing from the uniformization theorem. More generally, according to the Chern-Gauss-Bonnet formula, if
Minimal_volume
quasi-Fuchsian groups of the first kind is described by the simultaneous uniformization theorem of Bers. Fricke, Robert; Klein, Felix (1897), Vorlesungen über die
Quasi-Fuchsian_group
Topics referred to by the same term
domain name" One of the simply-connected Riemann surfaces – see uniformization theorem This disambiguation page lists articles associated with the title
Canonical_domain
Theorem concerning uniform convergence
In measure theory, an area of mathematics, Egorov's theorem establishes a condition for the uniform convergence of a pointwise convergent sequence of measurable
Egorov's_theorem
Doughnut-shaped surface of revolution
them that is both angle-preserving and orientation-preserving. The Uniformization theorem guarantees that every Riemann surface is conformally equivalent
Torus
Mathematical concept
Hunt's. This equivalency is sometimes given as definition for uniform integrability. Theorem 1: If ( X , M , μ ) {\displaystyle (X,{\mathfrak {M}},\mu )}
Uniform_integrability
Relation between sides of a right triangle
In mathematics, the Pythagorean theorem or Pythagoras's theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle
Pythagorean_theorem
Non-Euclidean geometry
according to the language of Riemann surfaces. According to the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic
Hyperbolic_space
French mathematician, physicist and engineer (1854–1912)
theory. He famously introduced the concept of the Poincaré recurrence theorem, which states that a state will eventually return arbitrarily close to
Henri_Poincaré
Polygon associated with a compact Riemann surface
determines the Riemann surface up to conformal equivalence. By the uniformization theorem, every compact Riemann surface has simply connected universal covering
Fundamental_polygon
Concept related to resolving singularities in algebraic geometry
at most 3. Local uniformization in positive characteristic seems to be much harder. Abhyankar (1956, 1966) proved local uniformization in all characteristics
Local_uniformization
Topological space that locally resembles Euclidean space
dimension 4: in low dimensions (2 and 3) it is geometric, via the uniformization theorem and the solution of the Poincaré conjecture, and in high dimension
Manifold
German mathematician (1826–1866)
the unit circle. The generalization of the theorem to Riemann surfaces is the famous uniformization theorem, which was proved in the 19th century by Henri
Bernhard_Riemann
Mathematical space
is the phenomenon that separates dimension 4 from others." The uniformization theorem for two-dimensional surfaces states that every simply connected
4-manifold
23 mathematical problems stated in 1900
linear differential equations having a prescribed monodromy group. 22. Uniformization of analytic relations by means of automorphic functions. 23. Further
Hilbert's_problems
Mathematical theorem
convergence theorem, named after the Italian mathematician Giuseppe Vitali, is a generalization of the better-known dominated convergence theorem of Henri
Vitali_convergence_theorem
Mathematical space
structure that can be associated with them. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected
3-manifold
One-dimensional complex manifold
complex analytic terms, the Poincaré–Koebe uniformization theorem (a generalization of the Riemann mapping theorem) states that every simply connected Riemann
Riemann_surface
Mathematical term in complex analysis
mapping theorem. More generally, if the spaces X and Y are Riemann surfaces, and Y is equipped with the metric coming from the uniformization theorem, then
Normal_family
Theorem in geometry
MR 1031909, S2CID 122626150 Morgan, John W. (1984), "On Thurston's uniformization theorem for three-dimensional manifolds", in Morgan, John W.; Bass, Hyman
Hyperbolization_theorem
Partial differential equation
a triumph of nineteenth-century geometry was the proof of the uniformization theorem, the analogous topological classification of smooth two-manifolds
Ricci_flow
Group representation of a Riemann surface
Lazarus Fuchs. By the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. More precisely this theorem states that a Riemann
Fuchsian_model
Mathematical folklore
In mathematical folklore, the "no free lunch" (NFL) theorem (sometimes pluralized) of David Wolpert and William Macready, alludes to the saying "no such
No_free_lunch_theorem
the sphere). The proof of Pu's inequality relies, in turn, on the uniformization theorem. In 2001, Sergei Ivanov presented another way to prove that the
Filling_area_conjecture
Algebraic variety in a projective space
lattice (also referred to as period lattice). According to the uniformization theorem already mentioned above, any torus of dimension 1 arises from an
Projective_variety
Approximation of a function by a polynomial
In calculus, Taylor's theorem gives an approximation of a k {\textstyle k} -times differentiable function around a given point by a polynomial of degree
Taylor's_theorem
Theorem in measure theory
In measure theory, Lebesgue's dominated convergence theorem gives a mild sufficient condition under which limits and integrals of a sequence of functions
Dominated_convergence_theorem
On uniformization of analytic relations
welcome and important preliminary studies. Koebe proved the general uniformization theorem that if a Riemann surface is homeomorphic to an open subset of the
Hilbert's twenty-second problem
Hilbert's_twenty-second_problem
Chinese-American mathematician (born 1949)
meshes), and in particular the computation of uniformizing maps as predicted by the uniformization theorem. In the case of genus-zero surfaces, a map is
Shing-Tung_Yau
Iterative method in conformal mapping
conditions: theses methods are then called Optimized Schwarz methods. Uniformization theorem Schwarzian derivative Schwarz triangle map Schwarz reflection principle
Schwarz_alternating_method
Invariant of simple closed curves
simple closed curve. According to the uniformization theorem, every domain has a conformal mapping to one of three uniform Riemann surfaces: an open unit disk
Loewner_energy
Riemannian manifold which satisfies vacuum Einstein equations
Einstein if and only if it has constant Gauss curvature. The classical uniformization theorem for Riemann surfaces guarantees that there is such a metric in every
Einstein_manifold
Theorem in vector calculus
theorem, also known as the Kelvin–Stokes theorem after Lord Kelvin and George Stokes, the fundamental theorem for curls, or simply the curl theorem,
Stokes'_theorem
Yau–Tian–Donaldson conjecture for Kähler–Einstein manifolds, and even the uniformization theorem. Gieseker stability Slope stability Bridgeland stability K-stability
Stability (algebraic geometry)
Stability_(algebraic_geometry)
Integer multiples of any irrational mod 1 are uniformly distributed on the circle
In mathematics, the equidistribution theorem is the statement that the sequence a, 2a, 3a, ... mod 1 is uniformly distributed on the circle R / Z {\displaystyle
Equidistribution_theorem
Algebraic curve in mathematics
{1}{4}}\right)}{2^{\frac {11}{8}}\pi ^{\frac {3}{4}}}}} Note that the uniformization theorem implies that every compact Riemann surface of genus one can be represented
Elliptic_curve
German mathematician (1932–2024)
classical mathematics – from the quadratic reciprocity law to the uniformization theorem, Kluwer 1991 Galois theory of p-extensions. Springer 2002 (older
Helmut_Koch
Theoretically optimal hypothesis test
is uniformly most powerful in these situations. Casella, G.; Berger, R.L. (2008), Statistical Inference, Brooks/Cole. ISBN 0-495-39187-5 (Theorem 8.3
Uniformly_most_powerful_test
Mathematical rule for inverting probabilities
Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes (/beɪz/), gives a mathematical rule for inverting conditional probabilities
Bayes'_theorem
American mathematician (born 1941)
particularly motivating mathematical theorem. The change was prompted by a special case of the uniformization theorem, according to which, in his own words:
Dennis_Sullivan
Sufficient criterion for uniform convergence
if the limit function is also continuous, then the convergence is uniform. The theorem is named after Ulisse Dini. If X {\displaystyle X} is a compact topological
Dini's_theorem
Counterintuitive result in probability
number of times. The theorem can be generalized to state that any infinite sequence of independent events whose probabilities are uniformly bounded below by
Infinite_monkey_theorem
Theorem in statistics
statistic is the unique uniformly minimum-variance unbiased estimator (UMVUE) of that quantity. The Lehmann–Scheffé theorem is named after Erich Leo
Lehmann–Scheffé_theorem
Relationship between derivatives and integrals
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at every
Fundamental theorem of calculus
Fundamental_theorem_of_calculus
Theorem in mathematics
In calculus and real analysis, the mean value theorem (or Lagrange's mean value theorem) is a theorem about differentiable functions, roughly stating
Mean_value_theorem
Type of mathematical group
surfaces of genus at least 2 are hyperbolic Riemann surfaces. Via the uniformization theorem this gives rise to a representation of its fundamental group in
Linear_group
Existence and uniqueness of solutions to initial value problems
known as Picard's existence theorem, the Cauchy–Lipschitz theorem, or the existence and uniqueness theorem. The theorem is named after Émile Picard,
Picard–Lindelöf_theorem
Parametrizes complex structures on a surface
are two simple examples that are immediately computed from the Uniformization theorem: there is a unique complex structure on the sphere S 2 {\displaystyle
Teichmüller_space
Complex numbers with non-negative imaginary part
Poincaré metric provides a hyperbolic metric on the space. The uniformization theorem for surfaces states that the upper half-plane is the universal covering
Upper_half-plane
greater) admit a metric of negative constant curvature (by the uniformization theorem), and the universal cover of the resulting Riemann surface is the
Small_cubicuboctahedron
Type of number sequence
sequence p(n) is uniformly distributed modulo 1. This was proven by Weyl and is an application of van der Corput's difference theorem. The sequence log(n)
Equidistributed_sequence
Generalization of closed graph, open mapping, and uniform boundedness theorem
analysis, the Ursescu theorem is a theorem that generalizes the closed graph theorem, the open mapping theorem, and the uniform boundedness principle
Ursescu_theorem
13 (1973), 31–47. doi:10.1215/kjm/1250523432 Ngaiming Mok. The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional
Frankel_conjecture
Latvian-American mathematician (1914–1993)
P. Steele Prize for mathematical exposition in 1975 for his paper "Uniformization, moduli, and Kleinian groups". In 1986, the New York Academy of Sciences
Lipman_Bers
Theorem in probability theory
variable, the theorem would be no longer valid. For example, let X n ∼ U n i f o r m ( 0 , 1 ) {\displaystyle X_{n}\sim {\rm {Uniform}}(0,1)} and Y n
Slutsky's_theorem
Describes the objects of a given type, up to some equivalence
and the geometrization conjecture – Three dimensional analogue of uniformization conjecture Berger classification – Concept in differential geometry
Classification_theorem
Curve from a cone intersecting a plane
(or 0), or x 2 − y 2 {\displaystyle x^{2}-y^{2}} . Indeed, by the uniformization theorem every surface can be taken to be globally (at every point) positively
Conic_section
American mathematician
Vol. 567, Springer, Berlin, 1977. John W. Morgan. On Thurston's uniformization theorem for three-dimensional manifolds. The Smith conjecture (New York
John_Morgan_(mathematician)
On convergent subsequences of functions that are locally of bounded total variation
In mathematics, Helly's selection theorem (also called the Helly selection principle) states that a uniformly bounded sequence of monotone real functions
Helly's_selection_theorem
Property of artificial neural networks
In the field of machine learning, the universal approximation theorems (UATs) state that neural networks with a certain structure can, in principle, approximate
Universal approximation theorem
Universal_approximation_theorem
Statement in probability theory
probability theory, Donsker's theorem (also known as Donsker's invariance principle, or the functional central limit theorem), named after Monroe D. Donsker
Donsker's_theorem
Pseudometric of complex manifolds
This gives many examples of hyperbolic complex curves, since the uniformization theorem shows that most complex curves (also called Riemann surfaces) have
Kobayashi_metric
Canadian mathematician
a geometrically meaningful part of the catenoid. By use of the uniformization theorem for surfaces with boundary, they were able to remove the condition
Ailana_Fraser
UNIFORMIZATION THEOREM
UNIFORMIZATION THEOREM
UNIFORMIZATION THEOREM
UNIFORMIZATION THEOREM
Girl/Female
Tamil
Nilakshi | நீலாகà¯à®·à¯€
Blue eyed
Girl/Female
Bengali, Indian, Telugu
Most Intelligent Person
Female
English
Variant spelling of English Tricia, TRISHA means "patrician, of noble descent."Â
Boy/Male
Hindu, Indian
Briliant
Girl/Female
Indian
Grass, Immortal one
Girl/Female
Hindu
A flower, Beautiful flowers, Cheerful, Pleased, Happy
Girl/Female
Hindu
Beloved of victory
Girl/Female
Hindu
Girl/Female
German, Swedish
Pearl
Boy/Male
Tamil
Sugapriyan | ஸà¯à®•à®¾à®ªà¯à®°à®¿à®¯à®¨
Wish to have peace
UNIFORMIZATION THEOREM
UNIFORMIZATION THEOREM
UNIFORMIZATION THEOREM
UNIFORMIZATION THEOREM
UNIFORMIZATION THEOREM
n.
A numerical coefficient in any particular case of the binomial theorem.
v. t.
To formulate into a theorem.
a.
Alt. of Theorematical
n.
The enunciation of a self-evident problem, in distinction from an axiom, which is the enunciation of a self-evident theorem.
a.
Theorematic.
n.
One who constructs theorems.
a.
Containing many names or terms; multinominal; as, the polynomial theorem.
a.
Of or pertaining to a theorem or theorems; comprised in a theorem; consisting of theorems.
n.
That which is considered and established as a principle; hence, sometimes, a rule.
n.
A statement of a principle to be demonstrated.
n.
A theorem or proposition so easy of demonstration as to be almost self-evident.