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	~~{{About\|Liouville's theorem on conformal mappings\|\|Liouville's theorem (disambiguation)}}~~		Hi there, I am Alyson Boon although it is not the name on my beginning certificate. North Carolina is exactly where we've been residing for many years and will by no means move. My day cheap psychic readings ([http://www.publicpledge.com/blogs/post/7034 www.publicpledge.com]) job is a travel agent. My husband doesn't like it the way I do but what I really like performing is caving but I don't have the time lately.<br><br>Also visit my web site: [http://chungmuroresidence.com/xe/reservation_branch2/152663 free psychic readings] ([http://cpacs.org/index.php?document_srl=90091&mid=board_zTGg26 cpacs.org])
	~~In [[mathematics]], '''Liouville's theorem''', proved by [[Joseph Liouville]] in [[#CITEREFMonge1850\|1850]]~~, is ~~a [[rigidity (mathematics)\|rigidity]] theorem about [[conformal mapping]]s in [[Euclidean space]]~~. ~~It states that any [[smooth function\|smooth]] conformal mapping on a domain of '''R'''<sup>''n''</sup>,~~ where '~~'n'' > 2, can be expressed as a composition of [[translation~~ (~~geometry)\|translations]],~~ [~~[similarity (geometry)\|similarities]], [[orthogonal matrix\|orthogonal transformations]] and [[inversive geometry\|inversions]]~~: ~~they are all [[Möbius transformation]]s~~. ~~This severely limits the variety of possible conformal mappings in '''R'''<sup>3<~~/~~sup> and higher-dimensional spaces. By contrast, conformal mappings in '''R'''<sup>2<~~/~~sup> can be much more complicated – for example, all [[simply connected]] planar domains are [[conformally equivalent]], by the [[Riemann mapping theorem]]~~.

	~~Generalizations of the theorem hold for transformations that are only [[weak derivative\|weakly differentiable]] {{harv\|Iwaniec\|Martin\|2001\|loc=Chapter 5}}~~. ~~The focus of such a study is the non-linear [[Cauchy–Riemann equations\|Cauchy–Riemann system~~]~~] that~~ is a ~~necessary and sufficient condition for a smooth mapping~~ '~~'&fnof;'' → Ω → '''R'''<sup>''n''</sup> to be conformal:~~
	~~:<math>Df^T Df = \left\|\det Df\right\|^{2/n}~~ I~~</math>~~
	~~where ''Df''~~ is ~~the [[Jacobian derivative]], ''T'' is the [[matrix transpose]], and ''~~I'~~' is~~ the ~~identity matrix~~. ~~A weak solution of this system is defined to be an element ''&fnof;'' of the [[Sobolev space]] ''W''{{su\|p=1,''n''\|b=loc}}(''Ω'','''R'''~~<~~sup~~>~~''n''~~<~~/sup~~>~~) with non-negative Jacobian determinant~~ [[almost everywhere]], such that the Cauchy–Riemann system holds at almost every point of Ω. Liouville's theorem is then that every weak solution (in this sense) is a Möbius transformation, meaning that it has the form
	:~~<math>f(x) = b + \frac{\alpha A (x-a)}{\|x-a\|^\epsilon}<~~/~~math>~~
	~~where ''a'',''b'' are vectors in '''R'''<sup>''n''<~~/~~sup>, α is a scalar, ''A'' is a rotation matrix, and ε = 0 or 2~~. Equivalently stated, any [[quasiconformal map]] of a domain in Euclidean space that is also conformal is a Möbius transformation. This equivalent statement justifies using the Sobolev space ''W''<sup>1,''n''</~~sup>, since ''&fnof;'' ∈ ''W''{{su\|p=1,''n''\|b=loc}}(''Ω'','''R'''<sup>''n''<~~/sup>) then follows from the geometrical condition of conformality and the ACL characterization of Sobolev space. The result is not optimal however: in even dimensions ''n'' = 2''k'', the theorem also holds for solutions that are only assumed to be in the space ''W''{{su\|p=1,''k''\|b=loc}}, and this result is sharp in the sense that there are weak solutions of the Cauchy–Riemann system in ''W''<sup>1,''p''</~~sup> for any ''p'' < ''k'' which are not Möbius transformations. In odd dimensions, it is known that ''W''<sup>1,''n''</sup> is not optimal, but a sharp result is not known.~~

	~~Similar rigidity results (in the smooth case) hold on any [[conformal manifold]]. The group of conformal isometries of an ''n''-dimensional conformal [[Riemannian manifold]~~] ~~always has dimension that cannot exceed that of the full conformal group SO~~(~~''n''+1,1). Equality of the two dimensions holds exactly when the conformal manifold is isometric with the~~ [[N sphere\|''n''-sphere]] or [[projective space]]. Local versions of the result also hold: The [[Lie algebra]] of [[conformal Killing field]]s in an open set has dimension less than or equal to that of the conformal group, with equality holding if and only if the open set is locally conformally flat.

	~~==References==~~
	* {{citation\|first=David E.\|last=Blair\|year=2000\|title=Inversion Theory and Conformal Mapping\|publisher=[[American Mathematical Society]]\|isbn=0-8218-2636-0\|chapter=Chapter 6: The Classical Proof of Liouville's Theorem\|pages=95–105}}.
	*{{citation\|first=Gaspard\|last=Monge\|authorlink=Gaspard Monge\|title=Application de l'analyse à la Géométrie\|year=1850\|publisher=Bachelier\|pages=609–616\|url=http://~~books~~.~~google~~.~~com/~~?id=~~iCEOAAAAQAAJ~~&dq=~~%22Application+de+l%27analyse+%C3%A0+la+g%C3%A9om%C3%A9trie%22}}~~
	*{{Citation \| last1=Iwaniec \| first1=Tadeusz \| author1-link=Tadeusz Iwaniec\|last2=Martin \| first2=Gaven \| title=Geometric function theory and non-linear analysis \| publisher=The Clarendon Press Oxford University Press \| series=Oxford Mathematical Monographs \| isbn=978-0-19-850929-5 \| id={{MathSciNet \| id = 1859913}} \| year=2001}}.
	*{{Citation \| last1=Kobayashi \| first1=Shoshichi \| title=Transformation groups in differential geometry \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| year=1972}}.
	*{{springer\|id=L/l059680\|title=Liouville theorems\|first=E.D.\|last=Solomentsev\|year=2001}}

	~~[[Category:Conformal mapping]]~~
	~~[[Category:Theorems in geometry]~~]

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{{About|Liouville's theorem on conformal mappings||Liouville's theorem (disambiguation)}}
In [[mathematics]], '''Liouville's theorem''', proved by [[Joseph Liouville]] in [[#CITEREFMonge1850|1850]], is a [[rigidity (mathematics)|rigidity]] theorem about [[conformal mapping]]s in [[Euclidean space]]. It states that any [[smooth function|smooth]] conformal mapping on a domain of '''R'''''n'', where ''n'' > 2, can be expressed as a composition of [[translation (geometry)|translations]], [[similarity (geometry)|similarities]], [[orthogonal matrix|orthogonal transformations]] and [[inversive geometry|inversions]]: they are all [[Möbius transformation]]s. This severely limits the variety of possible conformal mappings in '''R'''3 and higher-dimensional spaces. By contrast, conformal mappings in '''R'''2 can be much more complicated – for example, all [[simply connected]] planar domains are [[conformally equivalent]], by the [[Riemann mapping theorem]].

Generalizations of the theorem hold for transformations that are only [[weak derivative|weakly differentiable]] {{harv|Iwaniec|Martin|2001|loc=Chapter 5}}. The focus of such a study is the non-linear [[Cauchy–Riemann equations|Cauchy–Riemann system]] that is a necessary and sufficient condition for a smooth mapping ''&fnof;'' → Ω → '''R'''''n'' to be conformal:
:<math>Df^T Df = \left|\det Df\right|^{2/n} I</math>
where ''Df'' is the [[Jacobian derivative]], ''T'' is the [[matrix transpose]], and ''I'' is the identity matrix. A weak solution of this system is defined to be an element ''&fnof;'' of the [[Sobolev space]] ''W''{{su|p=1,''n''|b=loc}}(''Ω'','''R'''''n'') with non-negative Jacobian determinant [[almost everywhere]], such that the Cauchy–Riemann system holds at almost every point of Ω. Liouville's theorem is then that every weak solution (in this sense) is a Möbius transformation, meaning that it has the form
:<math>f(x) = b + \frac{\alpha A (x-a)}{|x-a|^\epsilon}</math>
where ''a'',''b'' are vectors in '''R'''''n'', α is a scalar, ''A'' is a rotation matrix, and ε = 0 or 2. Equivalently stated, any [[quasiconformal map]] of a domain in Euclidean space that is also conformal is a Möbius transformation. This equivalent statement justifies using the Sobolev space ''W''1,''n'', since ''&fnof;'' ∈ ''W''{{su|p=1,''n''|b=loc}}(''Ω'','''R'''''n'') then follows from the geometrical condition of conformality and the ACL characterization of Sobolev space. The result is not optimal however: in even dimensions ''n'' = 2''k'', the theorem also holds for solutions that are only assumed to be in the space ''W''{{su|p=1,''k''|b=loc}}, and this result is sharp in the sense that there are weak solutions of the Cauchy–Riemann system in ''W''1,''p'' for any ''p'' < ''k'' which are not Möbius transformations. In odd dimensions, it is known that ''W''1,''n'' is not optimal, but a sharp result is not known.

Similar rigidity results (in the smooth case) hold on any [[conformal manifold]]. The group of conformal isometries of an ''n''-dimensional conformal [[Riemannian manifold]] always has dimension that cannot exceed that of the full conformal group SO(''n''+1,1). Equality of the two dimensions holds exactly when the conformal manifold is isometric with the [[N sphere|''n''-sphere]] or [[projective space]]. Local versions of the result also hold: The [[Lie algebra]] of [[conformal Killing field]]s in an open set has dimension less than or equal to that of the conformal group, with equality holding if and only if the open set is locally conformally flat.

==References==
* {{citation|first=David E.|last=Blair|year=2000|title=Inversion Theory and Conformal Mapping|publisher=[[American Mathematical Society]]|isbn=0-8218-2636-0|chapter=Chapter 6: The Classical Proof of Liouville's Theorem|pages=95–105}}.
*{{citation|first=Gaspard|last=Monge|authorlink=Gaspard Monge|title=Application de l'analyse à la Géométrie|year=1850|publisher=Bachelier|pages=609–616|url=http://books.google.com/?id=iCEOAAAAQAAJ&dq=%22Application+de+l%27analyse+%C3%A0+la+g%C3%A9om%C3%A9trie%22}}
*{{Citation | last1=Iwaniec | first1=Tadeusz | author1-link=Tadeusz Iwaniec|last2=Martin | first2=Gaven | title=Geometric function theory and non-linear analysis | publisher=The Clarendon Press Oxford University Press | series=Oxford Mathematical Monographs | isbn=978-0-19-850929-5 | id={{MathSciNet | id = 1859913}} | year=2001}}.
*{{Citation | last1=Kobayashi | first1=Shoshichi | title=Transformation groups in differential geometry | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=1972}}.
*{{springer|id=L/l059680|title=Liouville theorems|first=E.D.|last=Solomentsev|year=2001}}

[[Category:Conformal mapping]]
[[Category:Theorems in geometry]]