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In the theory of several complex variables and complex manifolds in mathematics, a Stein manifold is a complex submanifold of the vector space of n complex dimensions. They were introduced by and named after Template:Harvs. A Stein space is similar to a Stein manifold but is allowed to have singularities. Stein spaces are the analogues of affine varieties or affine schemes in algebraic geometry.

Definition

A complex manifold $X$ of complex dimension $n$ is called a Stein manifold if the following conditions hold:

$X$ is holomorphically convex, i.e. for every compact subset $K \subset X$ , the so-called holomorphic convex hull,

\bar{K} = {z \in X : | f (z) | \leq \sup_{K} | f | \forall f \in 𝒪 (X)},

is again a compact subset of

X

. Here

𝒪 (X)

denotes the ring of holomorphic functions on

X

.

$X$ is holomorphically separable, i.e. if $x \neq y$ are two points in $X$ , then there is a holomorphic function

f \in 𝒪 (X)

such that

f (x) \neq f (y) .

Non-compact Riemann surfaces are Stein

Let X be a connected non-compact Riemann surface. A deep theorem of Behnke and Stein (1948) asserts that X is a Stein manifold.

Another result, attributed to Grauert and Röhrl (1956), states moreover that every holomorphic vector bundle on X is trivial.

In particular, every line bundle is trivial, so $H^{1} (X, 𝒪_{X}^{*}) = 0$ . The exponential sheaf sequence leads to the following exact sequence:

H^{1} (X, 𝒪_{X}) ⟶ H^{1} (X, 𝒪_{X}^{*}) ⟶ H^{2} (X, ℤ) ⟶ H^{2} (X, 𝒪_{X})

Now Cartan's theorem B shows that $H^{1} (X, 𝒪_{X}) = H^{2} (X, 𝒪_{X}) = 0$ , therefore $H^{2} (X, ℤ) = 0$ .

This is related to the solution of the Cousin problems, and more precisely to the second Cousin problem.

Properties and examples of Stein manifolds

The standard complex space $ℂ^{n}$ is a Stein manifold.
Every domain of holomorphy in $ℂ^{n}$ is a Stein manifold.
It can be shown quite easily that every closed complex submanifold of a Stein manifold is a Stein manifold, too.
The embedding theorem for Stein manifolds states the following: Every Stein manifold $X$ of complex dimension $n$ can be embedded into $ℂ^{2 n + 1}$ by a biholomorphic proper map.

These facts imply that a Stein manifold is a closed complex submanifold of complex space, whose complex structure is that of the ambient space (because the embedding is biholomorphic).

In one complex dimension the Stein condition can be simplified: a connected Riemann surface is a Stein manifold if and only if it is not compact. This can be proved using a version of the Runge theorem for Riemann surfaces, due to Behnke and Stein.
Every Stein manifold $X$ is holomorphically spreadable, i.e. for every point $x \in X$ , there are $n$ holomorphic functions defined on all of $X$ which form a local coordinate system when restricted to some open neighborhood of $x$ .
Being a Stein manifold is equivalent to being a (complex) strongly pseudoconvex manifold. The latter means that it has a strongly pseudoconvex (or plurisubharmonic) exhaustive function, i.e. a smooth real function $ψ$ on $X$ (which can be assumed to be a Morse function) with $i \partial \bar{\partial} ψ > 0$ , such that the subsets ${z \in X, ψ (z) \leq c}$ are compact in $X$ for every real number $c$ . This is a solution to the so-called Levi problem,^[1] named after E. E. Levi (1911). The function $ψ$ invites a generalization of Stein manifold to the idea of a corresponding class of compact complex manifolds with boundary called Stein domains. A Stein domain is the preimage ${z | - \infty \leq ψ (z) \leq c}$ . Some authors call such manifolds therefore strictly pseudoconvex manifolds.
Related to the previous item, another equivalent and more topological definition in complex dimension 2 is the following: a Stein surface is a complex surface X with a real-valued Morse function f on X such that, away from the critical points of f, the field of complex tangencies to the preimage X_c = f⁻¹(c) is a contact structure that induces an orientation on X_c agreeing with the usual orientation as the boundary of f⁻¹(−∞,c). That is, f⁻¹(−∞,c) is a Stein filling of X_c.

Numerous further characterizations of such manifolds exist, in particular capturing the property of their having "many" holomorphic functions taking values in the complex numbers. See for example Cartan's theorems A and B, relating to sheaf cohomology. The initial impetus was to have a description of the properties of the domain of definition of the (maximal) analytic continuation of an analytic function.

In the GAGA set of analogies, Stein manifolds correspond to affine varieties.

Stein manifolds are in some sense dual to the elliptic manifolds in complex analysis which admit "many" holomorphic functions from the complex numbers into themselves. It is known that a Stein manifold is elliptic if and only if it is fibrant in the sense of so-called "holomorphic homotopy theory".

Relation to smooth manifolds

Every compact smooth manifold of dimension 2n, which has only handles of index ≤ n, has a Stein structure provided n>2, and when n=2 the same holds provided the 2-handles are attached with certain framings (framing less than the Thurston-Bennequin framing).^[2]^[3] Every closed smooth 4-manifold is a union of two Stein 4-manifolds glued along their common boundary.^[4]

Notes

↑ PlanetMath: solution of the Levi problem
↑ Y. Eliashberg, Topological characterization of Stein manifolds of dimension > 2, Int. J. of Math. vol. 1, no 1 (1990) 29-46.
↑ R. Gompf, Handlebody construction of Stein surfaces, Ann. of Math. 148, (1998) 619-693.
↑ S. Akbulut and R. Matveyev, A convex decomposition for four-manifolds, IMRN, no.7 (1998) 371-381.

References

Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

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Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 (including a proof of Behnke-Stein and Grauert-Röhrl theorems)
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 (including a proof of the embedding theorem)
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 (definitions and constructions of Stein domains and manifolds in dimension 4)
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

[1] PlanetMath: solution of the Levi problem

[2] Y. Eliashberg, Topological characterization of Stein manifolds of dimension > 2, Int. J. of Math. vol. 1, no 1 (1990) 29-46.

[3] R. Gompf, Handlebody construction of Stein surfaces, Ann. of Math. 148, (1998) 619-693.

[4] S. Akbulut and R. Matveyev, A convex decomposition for four-manifolds, IMRN, no.7 (1998) 371-381.

[1]

[2]

[3]

[4]

@@ Line 1: / Line 1: @@
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+In the theory of [[several complex variables]] and [[complex manifold]]s in  mathematics, a '''Stein manifold''' is a complex [[submanifold]] of the [[vector space]] of ''n'' [[complex number|complex]] dimensions. They were introduced by and named after {{harvs|txt|authorlink=Karl Stein (mathematician)|first=Karl |last=Stein|year=1951}}. A '''Stein space''' is similar to a Stein manifold but is allowed to have singularities. Stein spaces are the analogues of affine varieties or affine schemes in algebraic geometry.
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+== Definition ==
+A [[complex manifold]] <math>X</math> of complex dimension <math>n</math> is called a '''Stein manifold''' if the following conditions hold:
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+* <math>X</math> is holomorphically convex, i.e. for every [[Compact space|compact]] subset <math>K \subset X</math>, the so-called ''holomorphic convex hull'',
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+::<math>\bar K = \{z \in X: |f(z)| \leq \sup_K |f| \ \forall f \in \mathcal O(X) \},</math>
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+:is again a ''compact'' subset of <math>X</math>. Here <math>\mathcal O(X)</math> denotes the ring of [[holomorphic]] functions on <math>X</math>.
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+* <math>X</math> is holomorphically separable, i.e. if <math>x \neq y</math> are two points in <math>X</math>, then there is a holomorphic function
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+:such that <math>f(x) \neq f(y).</math>
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+Let ''X'' be a connected non-compact Riemann surface. A deep theorem of [[Heinrich Behnke|Behnke]] and Stein (1948) asserts that ''X'' is a Stein manifold.
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+Another result, attributed to [[Hans Grauert|Grauert]] and [[Helmut Röhrl|Röhrl]] (1956), states moreover that every holomorphic vector bundle on ''X'' is trivial.
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+In particular, every line bundle is trivial, so <math>H^1(X, \mathcal O_X^*) =0 </math>. The [[exponential sheaf sequence]] leads to the following exact sequence:
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+: <math>H^1(X, \mathcal O_X) \longrightarrow H^1(X, \mathcal O_X^*) \longrightarrow H^2(X, \mathbb Z) \longrightarrow H^2(X, \mathcal O_X) </math>
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+Now [[Cartan's theorems A and B|Cartan's theorem B]] shows that <math>H^1(X, \mathcal O_X)= H^2(X, \mathcal O_X)=0 </math>, therefore <math>H^2(X, \mathbb Z)=0</math>.
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+This is related to the solution of the [[Cousin problems]], and more precisely to the second Cousin problem.
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+== Properties and examples of Stein manifolds ==
+* The standard complex space <math>\mathbb C^n</math> is a Stein manifold.
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+* Every [[domain of holomorphy]] in <math>\mathbb C^n</math> is a Stein manifold.
+* It can be shown quite easily that every closed complex submanifold of a Stein manifold is a Stein manifold, too.
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+* The embedding theorem for Stein manifolds states the following: Every Stein manifold <math>X</math> of complex dimension <math>n</math> can be embedded into <math>\mathbb C^{2 n+1}</math> by a [[biholomorphic]] [[proper map]].
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+These facts imply that a Stein manifold is a closed complex submanifold of complex space, whose complex structure is that of the [[ambient space]] (because the embedding is biholomorphic).
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+* In one complex dimension the Stein condition can be simplified: a connected [[Riemann surface]] is a Stein manifold [[if and only if]] it is not compact.  This can be proved using a version of the [[Runge theorem]] for Riemann surfaces, due to Behnke and Stein.
+* Every Stein manifold <math>X</math> is holomorphically spreadable, i.e. for every point <math>x \in X</math>, there are <math>n</math> holomorphic functions defined on all of <math>X</math> which form a local coordinate system when restricted to some open neighborhood of <math>x</math>.
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+* Being a Stein manifold is equivalent to being a (complex) ''strongly pseudoconvex manifold''. The latter means that it has a strongly pseudoconvex (or [[plurisubharmonic function|plurisubharmonic]]) exhaustive function, i.e. a smooth real function <math>\psi</math> on <math>X</math> (which can be assumed to be a [[Morse theory|Morse function]]) with <math>i \partial \bar \partial \psi >0</math>, such that the subsets <math>\{z \in X, \psi (z)\leq c \}</math> are compact in <math>X</math> for every real number <math>c</math>. This is a solution to the so-called '''Levi problem''',<ref>[http://planetmath.org/encyclopedia/LeviProblem.html PlanetMath: solution of the Levi problem]</ref> named after E. E. Levi (1911). The function <math>\psi</math> invites a generalization of ''Stein manifold'' to the idea of a corresponding class of compact complex manifolds with boundary called '''Stein domains'''. A Stein domain is the preimage <math>\{z|-\infty\leq\psi(z)\leq c\}</math>. Some authors call such manifolds therefore strictly pseudoconvex manifolds.
+*Related to the previous item, another equivalent and more topological definition in complex dimension 2 is the following: a Stein surface is a complex surface ''X'' with a real-valued Morse function ''f'' on ''X'' such that, away from the critical points of ''f'', the field of complex tangencies to the preimage ''X''<sub>''c''</sub> = ''f''<sup>&minus;1</sup>(''c'') is a [[Contact geometry|contact structure]] that induces an orientation on ''X<sub>c</sub>'' agreeing with the usual orientation as the boundary of ''f''<sup>&minus;1</sup>(&minus;∞,''c''). That is, ''f''<sup>&minus;1</sup>(&minus;∞,''c'') is a Stein [[Symplectic filling|filling]] of ''X<sub>c</sub>''.
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+Numerous further characterizations of such manifolds exist, in particular capturing the property of their having "many" [[holomorphic function]]s taking values in the complex numbers. See for example [[Cartan's theorems A and B]], relating to [[sheaf cohomology]]. The initial impetus was to have a description of the properties of the domain of definition of the (maximal) [[analytic continuation]] of an [[analytic function]].
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+In the [[GAGA]] set of analogies, Stein manifolds correspond to [[affine variety|affine varieties]].
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-== Christian Louboutin Outlet Uk  whom she turned her back on ==
+Stein manifolds are in some sense dual to the [[elliptic manifold]]s in complex analysis which admit "many" holomorphic functions from the complex numbers into themselves.  It is known that a Stein manifold is elliptic if and only if it is [[fibrant object|fibrant]] in the sense of so-called "holomorphic homotopy theory".
-Susannah Constantine<br><br>Her second book, entitled More to Wear, has won her a prestigious British Book Award and sold 670,000 copies.<br><br>Constantine was created right wealthy family; her father was successful in property and shipping sectors. She was privately educated since a child and proceeded up to now British royalty, David, Viscount Linley, during the 1980s. Constantine has become involved with fashion for years, originally getting work done in America for Giorgio Armani after which John Galliano within london. She met Trinny Woodall in 1994 who both proceeded to post a weekly fashion column, Willing to Wear. Following that these folks were commissioned to BBC Two to host the design and style series, More to use, from 2001 to 2005. She made regular appearances for a style advisor within the Oprah Show and following her success within the shows, she made to co host Trinny Susannah Undress. on ITV in the year 2006 and Undress the media in 2007. Constantine and Woodall have now dressed over 5,000 women.<br><br>This lady has co written many fashion advice books with Woodall, a few of which have grown to be best sellers in britain and United States. Constantine immediately declined the offer that has a four letter refusal and wrote "No fucking way" to the letter she'd received.[6] Her personal assistant, merely left to handle the situation, wrongly sent an original letter of invitation (containing Constantine's written note) back in the teachers. The institution then wrote instructions back, proclaiming that she is not welcome on the school.[3][4]Constantine originally did annually of Montessori training after she had left school but she then moved onto other projects[3] like studying sculpture in Brussels. She later said of your time which she "lost [her] virginity, went a bit mad."[7] Constantine had taught children for several years,[8] and even worked as a shop girl for Harrods.[3][9] She found the project, within their words, "boring" and in some cases tried shoplifting in the interest of excitement, but she's since expressed admiration for shop girls.[10] Constantine's love for horses had inspired her to become mounted policewoman in London, but was deterred when she found she was forced to do 4 years walking the beat.[10] She wrote a book about present giving, which prompted The Daily Telegraph to create a hostile article implying she'd never done a day's function in her lifetime, an issue that deeply upset her.[11] She's stated defiantly "I've always worked."[5]<br><br>Nancy now best known like a fashion guru and elegance advisor. Her fashion career started when she helped years like a shop girl in America for Giorgio Armani.[8] She later made a comeback to London with the aspiration of securing an occupation with Armani but found themselves doing work for designers including Richard James, Patrick Cox,[7] Alistair Blair and John Galliano which gained her a much greater comprehension of fashion, providing an intensive information about how the garment progresses from a sketch.[8] She then started dealing with the British Brain and Spine Foundation and so met the sports editor from the Daily Telegraph.[8] Whilst conducting a piece for GMTV, he asked Constantine to report the women's World Cup Final in cricket on discovering that it was her passion. It ensured they had further exposure in television and gained attention from Jane Root, controller of BBC Two, who took a risk and signed them up after their book venture along with their online business had failed badly.[9]<br><br>Constantine began co hosting What Not to make use of with Trinny Woodall in 2001, which required using her fashion advice and understanding to reform participants' appearances and fashion style. Constantine and Woodall hosted More to utilize until 2005 and have become better known for their tactile behaviour with the participants, direct advice,[15] and regularly speaking about breasts as tits which has become something of your trademark.[16] Never one to hold back, a notorious moment arose when Constantine spontaneously pulled ladies candidate's underwear down during filming as her knicker line was visible.[17] Critics with the items To not Wear argued the duo were too patronising with their subjects, an incident that they strongly disagree with.[7] Constantine declared that content come "from the heart",[18] and insists that the subjects "see you will find there's genuine appreciation for women. We enjoy women and they will notice that. Women just know."[7] She also stated "Ultimately, what we're doing is providing people with confidence. We're really the only can provide homeowners a feeling, who care how ordinary people dress. Not one person at Vogue magazine gives a shit. They deal with they, it's more creative and artistic " they're creating something beautiful. Nevertheless they don't cherish how their readers find yourself looking " whereas we do!"[7] The show made Constantine and Woodall household names and they're now known together as Trinny and Susannah. One reporter has simply referred to Constantine as "the one with 'big tits'."[16] Constantine and Woodall have even considered taking insurance out on their partnership should something unforeseen happen, although Constantine was about the move.[13]<br><br>Constantine offers the thought "anyone is capable of style. It doesn't matter what you do or where you're from."[16] She also claims that she finds dressing other women easy, but finds hard when dressing herself.[17] Constantine and Woodall share the fact that dressing to flatter shape is critical, stating "For us, its all about shape, and in what way which would cure a bodily defect. Were really like clothing doctors."[3]<br><br>During her time co hosting Much more to put on for five series, she and Woodall gained recognition for focus on the show, winning a Royal Television Society Award in 2002 if you are the top factual presenters.[19] In 2002, Constantine advised Jeremy Clarkson using a celebrity type of Much more to utilize.[20] After Clarkson appeared on the show, Nasir Khan stated "I'd rather eat my own personal hair than shop with these two [Constantine and Woodall] again".[16] Preparing its success, the show was nominated for your Features Award at the BAFTAS in the 2002 and 2003.[21] On the show Big Impression, impressionist Ronni Ancona latched onto spoofing Constantine's presenting techniques on which Not to Wear,[12][22] one among the shows on which Constantine's character has been spoofed.<br><br>For charity, in 2002 throughout the BBC's Children in need of help programme, both Constantine and Woodall performed their own form of Madonna's hit single "Vogue" that has a list of celebrity backing singers.[23] They became the faces of Nescaf in 2003 during which they were featured in television advertisements promoting business of coffee. In their contract, Constantine and Woodall gave a Nescaf competition winner a 10,000 makeover.[24]Following its ratings success, Much more to put on was promoted from BBC Two to BBC One in 2004, and secured even larger audiences. The format was changed slightly, with a One hour show with two makeovers rather than a 30 minute show with only one makeover[25] and also saw Constantine spending every day as one of her subjects. Much more to utilize soon gained her international fame if it was aired in countries including Spain and Portugal also in the American continent.<br><br>Constantine appeared on Children in need of funds in 2004, including a unique segment through which she gave the fictional EastEnders characters Little Mo and Mo Harris a makeover in the form of More make use of, commenting with them together with her usual "no nonsense" approach.[26] In 2005, Constantine voiced a robotic kind of herself while in the science fiction series, Doctor Who. Within the episode "Bad Wolf", Trine e and Zu Zana were two robots acting out a deadly futuristic kind of Much more make use of, offering a makeover to Jack Harkness inside a most gruesome form.[27]<br><br>The Oprah Winfrey Show in addition has seen Constantine and Woodall appearing regularly for a makeover and style experts giving fashion advice and guidelines in order to better overall appearances and giving numerous American women fashion makeovers.[28][29] In addition, they did an [http://christianlouboutinuk.mycylex.co.uk/ Christian Louboutin Outlet Uk] "Oprah bra and swimsuit intervention".[30] Reflecting on differences in women's style and willingness to submit to [http://cheapnikefreerunuk.mycylex.co.uk/ Cheap Nike Free Run Uk] makeovers between the UK and [http://uggbootsnz0.blog.co.nz/ Ugg Boots Nz] America, she stated that you will find very little difference, "The Americans are slightly less adventurous," knowning that "American women tend to be available to change and more receptive victims."[31]In 2006, Constantine and Woodall moved with the BBC to ITV within a deal reputedly worth 1.2 million[13] to get started on a whole new show, Trinny Susannah Undress., on 3 October. The other compilation of Trinny Susannah Undress. was transmitted in June 2007, but it maintained the format of series the one which saw Constantine and Woodall advising couples who have been finding problems into their marriage.[32][33]<br><br>Constantine think it is warming men and women could divulge heart's contents to them as confidantes, and even said "It's been fascinating to find out how looking good and achieving faith inside your appearance causes you to right sexier person, that makes you irresistible to your spouse again."[33] Constantine mentioned in an interview that filming the show was very emotionally draining, and for that reason, she often went home crying.[3] The programme could not come without its critics who questioned the depth at which Constantine and Woodall could cope with serious issues raised in the programme.[3] On 16 October 2006, they both appeared on NBC's The Today Show and performed makeovers on three women for the show.[34]<br><br>The brand new series on ITV saw a general change in format, confronting the leading fashion issues in the united kingdom.[35] Within a episode checking theme old, Constantine was [http://louisvuittonbeltuk.mycylex.co.uk/ Louis Vuitton Belt Uk] reconstructed as a seventy year old with the use of prosthetics and makeup.[36] She stated it took her four days to get rid of the view of herself aged so drastically, and compared the impression to using an "electric shock".[7] Broadcasting was scheduled for 7 November 2007 as well as the programme is newly entitled Trinny Susannah Undress The world. On 5 November and 28 December 2007 Constantine and Woodall appeared on Hi America and performed makeovers on three women with the show and gave style advice in line with the women's shapes.[37] In addition they reported for the fashion for the 80th Oscars specifically for the show in 2008.[38] Constantine and Woodall have dressed over 5,000 women by 2007.[39] With regards to their personal website, they state that their satisfaction remains with all the women they've got helped and inspired but not much while using success of book sales and viewing figures.<br><br>In 2003, when Constantine and Woodall were interviewed the very first time on Parkinson, actress Meg Ryan a controversial interview with Michael Parkinson, which generated negative publicity for Ryan. Parkinson declared he felt Ryan's behaviour to his fellow guests Constantine and Woodall, whom she turned her back on, was "unforgivable".[40]To coincide with Constantine's range from the BBC to ITV1 in 2006, she co launched an underwear range with Woodall called "Trinny and Susannah Magic Pants". Produced from nylon, they're designed to flatten the stomach, buttocks and thighs.[41] Constantine even wore them after giving birth to her third child.[42] She currently co writes an every week column to the Sun with Woodall.<ul>
+== Relation to smooth manifolds ==
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-== Nike Air Max Uk PETER OBETZ ==
+Every compact smooth manifold of dimension 2n, which has only handles of index ≤ n, has a Stein structure provided n>2, and when n=2 the same holds provided the 2-handles are attached with certain framings (framing less than the Thurston-Bennequin framing).<ref>[[Yakov Eliashberg|Y. Eliashberg]], Topological characterization of Stein manifolds of dimension > 2, Int. J. of Math. vol. 1, no 1 (1990) 29-46.</ref><ref>R. Gompf, Handlebody construction of Stein surfaces, Ann. of Math. 148, (1998) 619-693.</ref> Every closed smooth 4-manifold is a union of two Stein 4-manifolds glued along their common boundary.<ref>S. Akbulut and R. Matveyev, A convex decomposition for four-manifolds, IMRN,  no.7 (1998) 371-381.</ref>
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The graph showed Peter's percentages of survival as time passes.<br><br>"It started out at 90 % right then and there of the surgery and fell to fifteen percent after five years," Jeff says. "That was my scariest moment   there just a Fifteen percent chance which i was going to have my mate when camping a few years from now. The very next day, you'd caught your mom your little graph which the doctor had drawn and he or she wouldn't show it to you personally. And therefore I'd appear in, so you said, 'I i would love you to sit down and let me know everything. And so i did.' "<br><br>"I remember saying, 'I'm toast,' " Peter says.<br><br>"Your [http://mbtshoesnz0.blog.co.nz/ Mbt Nz] mom had said that to the surgeon. Choices said, 'No, he's lightly brown. He's not toast,' " Jeff says.<br><br>Cancer had been a wake call Peter says this individual have needed. "I was at employment where I had been miserable, also it provided me with the permission to leave," he admits that. "I went from creating a fortune to creating not much and being happier."<br><br>He sold his big house and gone to live in a condo. Brand new show after he moved, a unit downstairs popped and Jeff moved in. "The only time usually that you just live right on the doorstep for a good friend happens when you're a kid because often your following door neighbor is default your best friend," Peter says.<br><br>"Exactly. Second graders aren't that picky," Jeff says.<br><br>"When I believe of each and every area of my entire life: My marriage has changed, my job is different, where I live is different. Our friendship is truly the only thing that's constant," Peter says. "That's top-of-the-line gift you could've given me."<br><br>"When you are sick, everybody wished to say, 'Peter, I love you a whole lot, I'm so grateful for your friendship,' " Jeff says. "But I find myself so lucky that if anything may have became of you, there were never any ambiguity regarding how you feel about me or I feel with regards to you. Everyone loves you very, greatly."<br><br>"I accept you too, man."<br><br>Peter's been cancer free since 2009.<br><br>Audio produced for Morning Edition by Jud Esty Kendall. Today, a discussion between good friends. In 2004, Peter Obetz is at center of the divorce when he was clinically determined to have stage four esophageal cancer. This is when he called on Jeff Jarrett for help.<br><br>PETER OBETZ: Food would get stuck down my throat, also it got worse and worse, i really met with my doctor. I'd a tumor in this little esophagus wall. Anf the husband said, well, surgery has in regards to Ten percent death risk. And that i remember telling the doc, you mean to share with me, like, one guy on my own softball team is not going to allow it to become? You [http://cheapnikeshoesnz.blog.co.nz/ Cheap Nike Shoes Nz] already know, there's 10 guys on a team. He explained, yeah, that's just about it. And hubby said, we can [http://nikehighheelsuk.mycylex.co.uk/ Cheap Nike High Heels Uk] easily either do the surgery tomorrow, or we are able to wait till Tuesday.<br><br>JEFF JARRETT: Do you remember you called me?<br><br>OBETZ: Yeah.<br><br>JARRETT: I said you need to do it as being soon that you can. I spent the morning using your parents. And also the surgeon met along with us shortly after the surgery was completed and drew a graph. He was quoted saying, well, right here is the percentages of survival after some time. It began at Ninety percent make certain that with the surgery, and fell to fifteen percent after 5 years.<br><br>That had been my scariest moment, that there was simply a 15 percent chance i always would definitely have his mom by himself a few years from now. In the morning, you'd caught your mother achievable little graph the doctor had drawn, and she or he wouldn't show it to you. And so I'd are available in, so you said, I wish for you to sit down and identify everything. Therefore i did.<br><br>OBETZ: Going saying, I'm toast.<br><br>JARRETT: Your mom said that for the surgeon. Choices said, no. He's lightly brown. He is not toast.<br><br>OBETZ: You already know, to have gotten cancer, it was a wake call maybe which i needed. I used to be inside of a job where I became miserable, and yes it gave me the permission to exit. I went from creating a bundle to creating very little and being happier. I sold my big house, gone after a rental. Together with a week later, it downstairs exposed, therefore you moved in. My job has evolved. Home has evolved. Our friendship is often the only thing that's constant. That's top-of-the-line gift you could've given me.<ul>
+==Notes==
+<references/>
-   <li>[http://www.jamiatou.com/spip.php?article21 http://www.jamiatou.com/spip.php?article21]</li>
+== References ==
-   <li>[http://www.think-group.cn/VK/blog/article.php?type=blog&cid=2&itemid=1703208 http://www.think-group.cn/VK/blog/article.php?type=blog&cid=2&itemid=1703208]</li>
+* {{Citation | last1=Forster | first1=Otto | author1-link= Otto Forster | title=Lectures on Riemann surfaces | publisher=Springer Verlag | location=New-York | series=Graduate Text in Mathematics | isbn=0-387-90617-7 | year=1981 | volume=81}} (including a proof of Behnke-Stein and Grauert-Röhrl theorems)
+* {{Citation | last1=Hörmander | first1=Lars | author1-link= Lars Hörmander | title=An introduction to complex analysis in several variables | publisher=North-Holland Publishing Co. | location=Amsterdam | series=North-Holland Mathematical Library | isbn=978-0-444-88446-6 | mr=1045639  | year=1990 | volume=7}} (including a proof of the embedding theorem)
-   <li>[http://www.sebalo.info/spip/spip.php?article13 http://www.sebalo.info/spip/spip.php?article13]</li>
+* {{Citation | last1=Gompf | first1=Robert E. | title=Handlebody construction of Stein surfaces | mr=1668563  | year=1998 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=148 | issue=2 | pages=619–693 | doi=10.2307/121005 | jstor=121005 | publisher=The Annals of Mathematics, Vol. 148, No. 2}} (definitions and constructions of Stein domains and manifolds in dimension 4)
+*{{citation|mr=0580152|last1=Grauert|first1= Hans|last2= Remmert|first2= Reinhold|title=Theory of Stein spaces|series=Grundlehren der Mathematischen Wissenschaften |volume=236|publisher= Springer-Verlag|place= Berlin-New York|year= 1979|isbn= 3-540-90388-7 }}
-   <li>[http://www.imattidel93.com/forum/viewtopic.php?f=10&t=105424 http://www.imattidel93.com/forum/viewtopic.php?f=10&t=105424]</li>
+*{{citation|mr=0043219|last=Stein|first= Karl|title=Analytische Funktionen mehrerer komplexer Veränderlichen zu vorgegebenen Periodizitätsmoduln und das zweite Cousinsche Problem|language=German|journal=Math. Ann. |volume=123|year=1951|pages=201–222}}
+[[Category:Complex manifolds]]
-   <li>[http://woaishouji.com/forum.php?mod=viewthread&tid=69756&fromuid=33843 http://woaishouji.com/forum.php?mod=viewthread&tid=69756&fromuid=33843]</li>
- </ul>

Sylvester–Gallai theorem: Difference between revisions

Revision as of 23:20, 3 November 2013

Contents

Definition

Non-compact Riemann surfaces are Stein

Properties and examples of Stein manifolds

Relation to smooth manifolds

Notes

References

Navigation menu

Sylvester–Gallai theorem: Difference between revisions

Revision as of 23:20, 3 November 2013

Definition

Non-compact Riemann surfaces are Stein

Properties and examples of Stein manifolds

Relation to smooth manifolds

Notes

References

Navigation menu

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