User:Enyokoyama/sandbox

Mostow rigidity theorem

In mathematics, Mostow's rigidity theorem, or strong rigidity theorem, or Mostow–Prasad rigidity theorem, essentially states that the geometry of a finite-volume hyperbolic manifold of dimension greater than two is determined by the fundamental group and hence unique. The theorem was proven for closed manifolds by Mostow (1968) and extended to finite volume manifolds by Marden (1974) in 3-dimensions, and by Prasad (1973) in dimensions at least 3. Gromov (1981) gave an alternate proof using the Gromov norm.

Weil (1960, 1962) proved a closely related theorem, that implies in particular that cocompact discrete groups of isometries of hyperbolic space of dimension at least 3 have no non-trivial deformations.

While the theorem shows that the deformation space of (complete) hyperbolic structures on a finite volume hyperbolic n-manifold (for n > 2) is a point, for a hyperbolic surface of genus g > 1 there is a moduli space of dimension 6g − 6 that parameterizes all metrics of constant curvature (up to diffeomorphism), a fact essential for Teichmüller theory. In dimension three, there is a "non-rigidity" theorem due to Thurston called the hyperbolic Dehn surgery theorem; it allows one to deform hyperbolic structures on a finite volume manifold as long as changing homeomorphism type is allowed. In addition, there is a rich theory of deformation spaces of hyperbolic structures on infinite volume manifolds.

The original theorem

In the case of dimension n > 2 the theorem can be given in two equivalent ways, though seemingly different.

One statement of the Mostow rigidity theorem may be stated as:

Theorem 1: Suppose M and N are closed finite-volume hyperbolic n-manifolds of dimension n > 2. If ƒ : M  →  N is homotopy equivalent then ƒ : M  →  N is homotopic to a isometry from M to N.

Another statement is stated as:

Theorem 2: Suppose $\mathbb {H} /\Gamma =M$ and $\mathbb {H} /\Delta =N$ are closed hyperbolic manifolds of dimension n > 2. If $\Gamma$ and $\Delta$ are isomorphic then they are actually conjugate in $Isom(\mathbb {H} ^{n})$ .

In fact, the proof from Theorem 1 to Theorem 2 is shown by using the fact that if M and N are closed hyperbolic manifolds with isomorphic fundamental groups then M and N are homotopy equivalent. The inverse is shown by the fact that if ƒ : M  →  N is homotopic then f induces an isomorphism in the fundamental group. In the both of them the fundamental group is a important role.

In the case of n = 2 the rigidity in the above sense does not hold.

Relation to volume

Prasad extended Mostow’s results further by replacing the assumption that the manifolds be compact, with the assumption that they have finite volume. As such, the resulting theorem is sometimes known as Mostow-Prasad Rigidity.

Applications

The group of isometries of a finite-volume hyperbolic n-manifoldM (for n>2) is finite and isomorphic to Out(π₁(M)).

Mostow rigidity was also used by Thurston to prove the uniqueness of circle packing representations of triangulated planar graphs.

Proof of the simple estimate

${\begin{aligned}\left|\int _{C}f(x)e^{\lambda S(x)}dx\right|&\leqslant \int _{C}|f(x)|\left|e^{\lambda S(x)}\right|dx\\&\equiv \int _{C}|f(x)|e^{\lambda M}\left|e^{\lambda _{0}(S(x)-M)}e^{(\lambda -\lambda _{0})(S(x)-M)}\right|dx\\&\leqslant \int _{C}|f(x)|e^{\lambda M}\left|e^{\lambda _{0}(S(x)-M)}\right|dx&&\left|e^{(\lambda -\lambda _{0})(S(x)-M)}\right|\leqslant 1\\&=\underbrace {e^{-\lambda _{0}M}\int _{C}\left|f(x)e^{\lambda _{0}S(x)}\right|dx} _{\text{const}}\cdot e^{\lambda M}.\end{aligned}}$

References

• Gromov, Michael (1981), "Hyperbolic manifolds (according to Thurston and Jørgensen)", Bourbaki Seminar, Vol. 1979/80, Lecture Notes in Math., vol. 842, Berlin, New York: Springer-Verlag, pp. 40–53, doi:10.1007/BFb0089927, ISBN 978-3-540-10292-2, MR 0636516
• Marden, Albert (1974), "The geometry of finitely generated kleinian groups", Annals of Mathematics. Second Series, 99 (3): 383–462, doi:10.2307/1971059, ISSN 0003-486X, JSTOR 1971059, MR 0349992, Zbl 0282.30014
• G. D. Mostow, Quasi-conformal mappings in n-space and the rigidity of the hyperbolic space forms, Publ. Math. IHES 34 (1968) 53–104.
• Mostow, G. D. (1973), Strong rigidity of locally symmetric spaces, Annals of mathematics studies, vol. 78, Princeton University Press, ISBN 978-0-691-08136-6, MR 0385004
• Prasad, Gopal (1973), "Strong rigidity of Q-rank 1 lattices", Inventiones Mathematicae, 21 (4): 255–286, doi:10.1007/BF01418789, ISSN 0020-9910, MR 0385005, S2CID 55739204
• R. J. Spatzier, Harmonic Analysis in Rigidity Theory, (1993) pp. 153–205, appearing in Ergodic Theory and its Connection with Harmonic Analysis, Proceedings of the 1993 Alexandria Conference, Karl. E. Petersen, Ibrahim A. Salama, eds. Cambridge University Press (1995) ISBN 0-521-45999-0. (Provides a survey of a large variety of rigidity theorems, including those concerning Lie groups, algebraic groups and dynamics of flows. Includes 230 references.)
• William Thurston, The geometry and topology of 3-manifolds, Princeton lecture notes (1978–1981). (Gives two proofs: one similar to Mostow's original proof, and another based on the Gromov norm)
• Weil, André (1960), "On discrete subgroups of Lie groups", Annals of Mathematics. Second Series, 72 (2): 369–384, doi:10.2307/1970140, ISSN 0003-486X, JSTOR 1970140, MR 0137792
• Weil, André (1962), "On discrete subgroups of Lie groups. II", Annals of Mathematics. Second Series, 75 (3): 578–602, doi:10.2307/1970212, ISSN 0003-486X, JSTOR 1970212, MR 0137793

BV formalism

In theoretical physics, the Batalin–Vilkovisky (BV) formalism (named for Igor Batalin and Grigori Vilkovisky) was developed as a method for determining the ghost structure for Lagrangian gauge theories, such as gravity and supergravity, whose corresponding Hamiltonian formulation has constraints not related to a Lie algebra (i.e., the role of Lie algebra structure constants are played by more general structure functions). The BV formalism, based on an action that contains both fields and "antifields", can be thought of as a vast generalization of the original BRST formalism for pure Yang–Mills theory to an arbitrary Lagrangian gauge theory. Other names for the Batalin–Vilkovisky formalism are field-antifield formalism, Lagrangian BRST formalism, or BV-BRST formalism. It should not be confused with the Batalin–Fradkin–Vilkovisky (BFV) formalism, which is the Hamiltonian counterpart.

• improve BFV-frmalism(Hamiltonian formalism) is earlier than BV formalism, in usual BV formalism is used.
• AKSZ construction uniformization of them.

A-polynomial

In mathematics A-polynomial is an invariant for knots derived from the fundamental group of a given knot. It is closely related to hyperbolic volume and to Mahler measure. It was firstly defined by .

AJ conjecture states that a certain polynomial determined by the colored Jones polynomial would be in fact the A-polynomial.

definition

Let K be a knot in S³ and M the exterior of K. The boundary of M is a torus, which has the meridian-longitude basis (μ,λ) and so its fundamental group is $\pi _{1}(\partial M)=\mathbb {Z} ^{2}$ . Consider a representation $\rho :\pi _{1}(M)\rightarrow SL(2,\mathbb {C} )$ . Any such representation $\rho$ will be conjugate to one that is upper triangular, so we will restrict to upper triangular representations. For (μ,λ), the matrices

$\rho (\mu )={\begin{pmatrix}m&t\\0&m^{-1}\end{pmatrix}}\ ,\ \ \ \ \rho (\lambda )={\begin{pmatrix}l&k\\0&l^{-1}\end{pmatrix}}$

can be corresponded, where t,k,l and m are complex numbers.

The set of all such points (l,m), say S, consists of several components of various dimensions. Take a component $C$, and consider its Zariski closure ${\overline {C}}$ . If ${\overline {C}}$ is the zero set of only a single polynomial, call this polynomial $F_{C}$ . Define the product of all such polynomials $F_{C}$ to be A' . Finally any polynomial found this way will be divisible by (l-1) , which come from the abelianization of the representations of $\rho$ . Thus we will divide by (l-1) to obtain the A-polynomial $A_{K}(l,m)$ .

properties

The A-polynomials have some interesting properties. For example,

1. If K is the unknot, $A_{K}(l,m)=\pm 1\ .$
2. $A_{K}(l,m)=A_{K}(l^{-1},m^{-1})$, up to multiplication by powers of l and m.
3. Under the change of basis

${\begin{pmatrix}\gamma _{l}\\\gamma _{m}\end{pmatrix}}\longrightarrow {\begin{pmatrix}a&b\\c&d\end{pmatrix}}{\begin{pmatrix}\gamma _{l}\\\gamma _{m}\end{pmatrix}},\ \ \ \ {\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in SL(2,\mathbb {Z} )$

The A-polynomial transforms as

$A(l,m)\longrightarrow A(l^{d}m^{-b},l^{-c}m^{a})\ .$

4. If $K_{1}$ and $K_{2}$ are two knots and $K_{1}\#K_{2}$ is their connected sum then $A_{K_{1}\#K_{2}}$ is divisible by $A_{K_{1}}\cdot A_{K_{2}}/(l-1)$ .

5. A-polynomial can distinguish mirror knots.

6. A-polynomial has integer coefficients.

These are important for knot theory.

AJ conjecture

AJ conjecture is important for knot theory because if this conjecture would be true then the (colored) Jones polynomial distinguishes unknot.

application

It describes some topics such as A-polynomials, Melvin-Morton conjecture, and BTZ black holes etc., from physical viewpoint.

Kontsevich integral

Definition

Let K to be a Morse knot embedded into 3-dimensional space C× R. Namely, if we denote the map from S¹ to C× R as s → (z (s), h (s)) then every critical points of h is isolated. Further, one of critical points of h has a inverse point that consist of one point.

definetion by integration

Kontsevich invarinat or Kontsevich integral of a knot K is the infinte series Z (K) defined by the following relation:

$Z(K)=\sum _{m=0}^{\infty }{\frac {1}{(2\pi {\sqrt {-1}})^{m}}}\int _{-\infty <t_{1}<t_{2}<\dots <t_{m}<\infty }\sum _{p\in P}(-1)^{\sharp p\downarrow }D_{p}\wedge _{i=1}^{m}{\frac {dz_{i}-dz_{i}^{\prime }}{z_{i}-z_{i}^{\prime }}}$

where

• C×{ t_i } と K の共通部分から二点 z_i と z' _i を選んで組にする。このような組の列 {(z_i , z' _i)}_{i = 1,2,...,m} 全てからなる集合が P。
• #p↓ は p に現れる 2m 個の点のうち、そこで K が下向きになっているものの個数である。
• D_p is the chord diagram obtained from the inverse image of each points p = (z_i , z' _i ) .

The differential form in the right hand side comes from Knizhnik–Zamolodchikov equations. Knizhnik–Zamolodchikov equations determine the flat connection in donfigration space in which the integral along the loop in the configuration space, i.e. along the braie, reserves its value under the infinitesimal deformation. This leads the Kontsevich integral to a invariant.

definition by combinatorics

Suppose that K can be decomposed into some factor, C×{t_i } and on the sectional surface the section of K may be on the real axis. Then K is considered to be constructed by the compositions and the tensor products of tangles as above.

For the elementary component consisting of K Kontsevich invariant is defined as follows:

• Z () = ()·e^t/2, Z () = ()·e^-t/2。ここで t は水平な一本のコードだけを持つコード図で、e^x は形式的な指数写像。
• Z() = $\sharp$ U^-1/2, Z() = $\sharp$ U^-1/2。ここで U は極大点と極小点をそれぞれ二つもつ自明な結び目のコンツェビッチ不変量で $\sharp$ は連結和。
• Z() は直接コンツェビッチ積分を計算することで得られる。この値を Φ と表記すると、 Z() = Φ^-1。

そして、合成とテンソル積に対しては以下のようにコンツェビッチ不変量を定める。

• Z(s·u)=Z(s)·Z(u)。
• Z(s ⊗ u)=Z(s) ⊗ Z(u)。

Unlike the usual tangle, 通常のタングルとは異なり、隣り合う端点との距離が等しいことを仮定しないことに注意すべきである(これにより、ここで扱うようなタングルを非結合的タングル、準タングルと呼ぶこともある)。準タングルはモノイド圏を成すが、モノイド積に関して (a ⊗ b)⊗ c = a ⊗ (b ⊗ c) は成立しない。Φ はこの両辺の間の同型を与え、五角関係式(モノイド圏のコヒーレンス条件)をみたす。Φ(またはリー代数由来のウェイトシステムによる像)をドリンフェルト・アソシエータ と呼ぶこともある。上記の U や Φ は無限級数であり、一般の結び目に対する Z の値を求めることは低次の項を除いて非常に難しい。

Propeties

• Since there is only one kind in the Jacobi diagram with degree 0 the part of degree 0 of Kontsevich invariant is invarint on intersection transformation for knots. Therefore, the coefficients of Kontsevich invarinats are of finite type.
  • In particular, the coefficentes of degree 2 are essentially an Alexander-Conwey polynomial.
• The values of Kontsevich invarinats form a group: denote coproduct as Δ , they satisfy Δ(Z ( K )) =Z ( K ) ⊗ Z ( K ) . Then there is a element z( K ) of $A(S^{1})$ that is written as Z ( K ) = exp (z (K )) . All of z ( K ) in the chord diagram have some "legs" for connection to some connected loops $S^{1}$ , which are called loop expansions.
• Kontsevich invarinats are conjectured to be complete invariants for knots.

universality for invarinats of difinite type

次数 m の有限型不変量 v から m 次のヤコビ図に対するウェイトシステム W_v を構成することができ、一方ウェイトシステム W に対して、 W·Z の m 次の係数は m 次の有限型不変量である。コンツェビッチ不変量は m 次の有限型不変量の空間と m 次のヤコビ図に対するウェイトシステムの空間の間の同型対応を与える(実際には商空間の間の同型となる。)。

• sl₂ から定まるウェイトシステムからはジョーンズ多項式の係数、sl_n の場合はホンフリー多項式の係数が導かれる。

歴史

コンツェビッチ不変量はまずコンツェビッチによって反復積分の形で定義された。しかしその定義から、結び目を水平線で幾つかの部分に分割し、部分ごとに不変量の値を求めてもよいことが容易にわかる。実際、レ(Le) と村上順は、結び目の生成系であるタングルを準タングルに拡張し、生成元ごとにコンツェビッチ不変量の値を計算することで組み合わせ的な定義を得た。同時に彼らは紐のねじれ(framing)に対応するコンツェビッチ不変量の値も定式化し、三次元多様体に対する普遍量子不変量への道を開いた(技術的な要請から、反復積分による定義ではヤコビ図(正確にはコード図)に FI 関係式が必要で、紐のねじれの情報は値に反映されなかった)。

Kontsevich invariant is essentially infinite series it is too difficult to decide its value. The value for an unknot is determined in .

External links

• Chirka, E. M. (2001) [1994], "Hartogs theorem", Encyclopedia of Mathematics, EMS Press
• "Failure of Hartogs' theorem in one dimension (counterexample)". PlanetMath.
• Hartogs' theorem at PlanetMath.
• "Proof of Hartogs' theorem". PlanetMath.

Category;Several complex variables Category;Theorems in complex analysis

An example on analytic continuation

As described in the previous there are similar results in several variables case as one variable case. However, there are very different aspects in several variable case. For example, Riemann mapping theorem, Mittag-Leffler's theorem, Weierstrass theorem, Runge's theorem and so on can not apply to the several variables case as it is in one variable case. The following example of analytic continuation in two variables shows these differences, which was one of motivations to complex analysis in several variables.

In several variables analytic continuation is defined in the same way as in one variable case. Namely, let $U,V$ be open subsets in $\mathbb {C} ^{n}$, $f\in {\mathcal {O}}(U)$ and $g\in {\mathcal {O}}(V)$. Assume that $U\cap V\neq \phi$ and $W$ is a connected component of $U\cap V$. If $f|_{W}=g|_{W}$ then $h$ is defined as

$h(z)={\begin{cases}f(z)&z\in U,\\g(z)&z\in V.\end{cases}}$

The above $h$ is called analytic continuation of $f$ or $g$. Note that $h$ is uniquely determined by the identity theorem but may be multi-valued.

In one variable case, $n=1$, for any open domain $U\varsubsetneqq \mathbb {C}$ there is a holomorphic function $f$ on $U$ such that cannot analytically continued beyond $U$. That is, for any $a\in \partial U$, $f={\frac {1}{z-a}}$ cannot be analytically continued beyond $a$. However, in several variables case, $n\geq 2$, it would occur that there are a restrictly larger open domain ${\widetilde {U}}\varsupsetneqq U$ such that all $f\in {\mathcal {O}}(U)$ can be continued analytically to ${\tilde {f}}\in {\widetilde {U}}$. This phenomenon is called Hartogs' phenomenon, which cannot occur in one variable case.

This phoenomenon lead the vast influence for some area of mathematics.

1. Hartogs himself tried to provide a proof of Hartogs extension theorem that in the way of polydisks.
2. Hoermandoer's method is of the ${\overline {\partial }}$-equation.
3. the idea of integral kernel, i.e. Bergmann kernel etc. .

Domain for Holomorphy

Main article: Domain of holomorphy

Riemann mapping theorem uniformaization theorem classification with curvature --> Calabi conjecture --> the problem of unique decision for the Kahler-Einstein manifold in the space with positive curvature in general dimension.= complex Monge-Ampere equation

In one variable case Runge's theorem states that in a compact domain D in complex plane all the holomorphic function on D is approximated by some sequences of polynomials or rational functions. But in several variables this result does not hold. For example,

the extension of Runge's theorem and Weierstrass approximation theorem by Mergelyan -> Mergelyan's theorem Mergelyan's theorem stated the following:

Let K be a compact subset of the complex plane C such that C\K is connected. Then, every continuous function f : K$\to$ C, such that the restriction f to int(K) is holomorphic, can be approximated uniformly on K with polynomials. Here, int(K) denotes the interior of K.

In one variable there can exists holomorphic functions on any connected open C, but In several case this does not always hold. Only on the real values included in the original complex domain U there is the definition of plurisubharmonic functions

for example

psuedo convex domain <--> domain of holomorphy ? Levi's problem

with a bravity, pseudo convex domain is the domain the first cousin problem is always solved

the second cousin problem is always solved if and only if H^2(M, Z)=0.

we can define a holomorphic function with zeros accumulating everywhere on the boundary of the domain, which must then be a natural boundary for a domain of definition of its inverse. For n \geq 2 this is no longer true, as it follows from Hartogs' lemma.

In one variable case Mittag-Leffler's theorem and Weierstrass' theorem are problems how one determines the rational function with given zero's and poles.

Using Riemann-Hurwitz formula

Using Riemann-Hurwitz formula the hyperelliptic curve with genus g is one defined by a equation with degree n = 2g + 2. Suppose the bijective morphism f : X → P¹ with ramification degree 2, where X is a curve with genus g and P¹ is the Riemann sphere. Let g₁ = g and g₀ be the genus of P¹, the Riemann-Hurwitz formula brings

$2-2g_{1}=2(1-g_{0})-\sum _{s\in X}(e_{s}-1)$

where s is over all ramified point on X. The number of tamified points is finite, n, so n = 2g + 2.

Clifford's Theorem: Let D be an effective special divisor in the curve X. Then

${\textrm {dim}}D={\frac {1}{2}}{\textrm {deg}}D$

occurs if and only if either D = 0, or D = K(canonical divisor), or X is hyperelliptic and D is a multiple of the unique g₂¹ on X, where g_d^r stands for a linear system of dimension r and degree d.

other known results

All curves of genus 2 are hyperelliptic, but for genus ≥ 3 the generic curve might not be hyperelliptic. More results is rather known on the generalized hyperelliptic i.e. containing nonhyperelliptic curve, called superelliptic curves, than the elliptic cureves. One geometric characterization of hyperelliptic curves is via Weierstrass points. More detailed geometry of non-hyperelliptic curves is read from the theory of canonical curves, the canonical mapping being 2-to-1 on hyperelliptic curves but 1-to-1 otherwise for g > 2. Trigonal curves are those that correspond to taking a cube root, rather than a square root, of a polynomial.

The definition by quadratic extensions of the rational function field works for fields in general except in characteristic 2; in all cases the geometric definition as a ramified double cover of the projective line is available, if it is assumed to be separable.

Hyperelliptic curves also appear composing entire connected components of certain strata of the moduli space of Abelian differentials.

Hyperelliptic curves can be used in hyperelliptic curve cryptography for cryptosystems based on the discrete logarithm problem.