Hamiltonian systems, symplectic flows, classical integrable systems

**0**

votes

**0**answers

131 views

### SLAGs on elliptic curves are only lines?

Let $E=\mathbb{C}/(\mathbb{Z}+\tau\mathbb{Z})$ be an alliptic curve. Define a holomorphic 1-form $dz=dx+idy$ and a Kahler form $\omega=dx\wedge dy$. How can one prove that special Lagrangians on $E$ ...

**3**

votes

**1**answer

177 views

### Trivial canonical bundle of a Ricci-flat, simplyconnected Kähler manifold

Hallo,
I have two questions where I do not really know how to deal with them. Let $(M,J,g)$ be a Kähler manifold, where $g$ is the Riemannian metric and denote by $\omega(\cdot , \cdot) = g(J \cdot ...

**7**

votes

**1**answer

257 views

### quasi conformal, area preserving homomorphisms of the disc

Restricting a quasi-conformal homeomorphism of the disc to the boundary gives a surjective homomorphism from $QC(D^2)$ (quasi-conformal homeos of $D^2$) to $QS(S^1)$ (quasi-symmetric homeos of the ...

**1**

vote

**0**answers

168 views

### A question on the SYZ mirror symmetry

A toy SYZ mirror symmetry is described as follows. Let $X$ be a $n$-dimensional compact manifold. The contangent bundle $T^*X$ has a natural symplectic structure locally given by the $2$-form ...

**8**

votes

**3**answers

963 views

### Intuition for Levi-Civita connection via Hamiltonian flows

A Euclidean metric on a manifold $X$ defines a function on the symplectic space $T^*X$ whose Hamiltonian flow gives geodesics. Is there a similar interpretation of the Levi-Civita connection?

**9**

votes

**1**answer

719 views

### What is known about the strong Arnold conjecture?

Here are the two versions of Arnold's conjecture on Hamiltonian orbits:
Let $(M,\omega)$ be a closed symplectic manifold, and let $H: \mathbb{R/Z} \times M \to \mathbb{R}$ be a nondegenerate ...

**0**

votes

**0**answers

85 views

### Ricci-flat Kähler metrics on symmetric varieties

Hallo,
I have a question on a paper of Azad and Kobayashi "Ricci-flat Kähler metrics on symmetric varieties". Here is the link: ...

**1**

vote

**1**answer

214 views

### Torsion-free $G$-Structures

I have the following question. Let $G \subset SO(n)$ be a Lie Group and $M$ be a smooth manifold of dimension $n$. Furthermore let $P$ be a $G$-structure on $M$ i.e. $P$ is a principal subbundle of ...

**8**

votes

**0**answers

226 views

### Homology classes represented by $J$-holomorphic curves

Let $\Sigma$ be a compact Riemann surface with complex structure $j$. Let $(M,J)$ be an almost complex manifold. A map $u: \Sigma \rightarrow M$ is called $J$-holomorphic if
$$ du \circ j = J \circ ...

**3**

votes

**1**answer

352 views

### conservation law and generalized Symplectic Monge-Ampere equation arising from 3-variables

If we have a Jacobi PDE system with conservation law $\theta \in \Omega^1(M)$ such that $d \theta$ is non-degenerate 2-form , then we know this fact that it can be written as symplectic 2D ...

**6**

votes

**1**answer

534 views

### Darboux like theorem for non-degenerate 3-forms in 6-manifolds

we know Darboux theorem for higher-symplectic geometry is not correct in general,
but is there any Darboux like theorem for non-degenerate 3-forms in 6-manifolds?

**0**

votes

**1**answer

197 views

### $SU(n)$-structures on a manifold

I have the following question. Consider a $2n$-dimensional almost complex manifold $M$. Assume that on $M$ there exists a complex valued $n$-form $\Omega$ and a $2$-form $\omega$ such that:
$\Omega$ ...

**5**

votes

**1**answer

325 views

### Qustions on R.Bryant's papaer “Calibrated embeddings in the special Lagrangian and coassociative cases”

I am reading the paper "Calibrated embeddings in the special Lagrangian and coassociative cases" by R.Bryant (here the link: http://arxiv.org/abs/math/9912246) and there are certain things that are ...

**0**

votes

**2**answers

259 views

### Hamiltonian group actions - examples

I want to study about Symplectic group actions and moment map, especially Hamiltonian Group Actions. Can you help me with some concretely example of Hamiltonian group actions ? Where can I find some ...

**1**

vote

**1**answer

141 views

### finding effective 2-form corresponding to an equation

What is the effective 2-form corresponding to the equation
$det Hess v=(v-q_1v_{q_1}-q_2v_{q_2})^4$
you can find the definition of effective forms here

**4**

votes

**1**answer

368 views

### What is the physical interpretation of the canonical relations?

This question arises when studying semi-classical analysis. Since Weinstein's creed claims that "everything is Lagrangian", where a point in the phase space of classical mechanics is just a cotangent ...

**3**

votes

**1**answer

518 views

### Symplectic block-diagonalization of a real symmetric Hamiltonian matrix

Given a $2n\times 2n$ real, symmetric, Hamiltonian matrix $W$ (anticommutes with the symplectic metric), is there an orthogonal, symplectic matrix $R$ such that $R^\top WR$ is block-diagonal?
Being ...

**1**

vote

**3**answers

422 views

### Reference on generators of subgroups of symplectic groups

We should start with the definition of the symplectic group for an arbitrary ring $R$.
The symplectic group $Sp(g,R)$ is the subgroup of $SL(2g,R)$ such that all elements satisfy $M=J_g^t M J_g$ with ...

**7**

votes

**1**answer

294 views

### How many “elementary” characterizations of twisted SU(2) representation varieties are known?

If $\Sigma_g$ is a genus-$g$ surface, $g \geq 2$, then let $\mathcal{M}(\Sigma_g)$ be its twisted SU(2) representation variety, i.e. $$\mathcal{M}(\Sigma_g) := \{ (A_1, B_1, \ldots, A_g, B_g) \in ...

**6**

votes

**1**answer

306 views

### Fundamental groups of symplectic leaves

Let $X = \operatorname{Spec}(R)$ be a conical Poisson variety over $\mathbb{C}$. This means
that $R$ is a non-negatively graded Poisson algebra over $\mathbb{C}$
with $R_0 = \mathbb{C}$ and that the ...

**0**

votes

**1**answer

215 views

### special Lagrangian n-Torus has Tubular neighbourhood?

Let $\imath :T^{n}\rightarrow X$ is a special Lagrangian n-Torus so that $\imath(T^{n})=L$ and all small special Lagrangian deformations of $L$ are flat then why $L$ has Tubular neighbourhood which ...

**1**

vote

**0**answers

173 views

### Question in the paper of Robert Bryant “Calibrated embeddings in the special Lagrangian and coassociative cases”

Hallo,
I am trying to read the paper "Calibrated embeddings in the special Lagrangian and coassociative cases" by Robert Bryant and I have a question. I hope that maybe one of you could give me some ...

**1**

vote

**1**answer

97 views

### Is the space of half-dimensional symplectic linear subspaces of $\mathbb{R}^{4n}$ which trivially intersect $\mathbb{R}^{2n}\times\{0\}$ contractible?

I would like to know whether the subspace of the symplectic Grassmanian $Gr_{2n}^{Sp}(\mathbb{R}^{4n})$ consisting of symplectic linear subspaces in $\mathbb{R}^{4n}$ which have dimension $2n$ and ...

**5**

votes

**3**answers

777 views

### what prevents a manifold to be symplectic?

Are there any obstructions known which prevent an even dimensional orientable manifold from
being symplectic? I am a novice in this area so I unfortunately I cannot make the question more precise. ...

**1**

vote

**0**answers

52 views

### action aifference for cylinders in a symplectic cobordism

Suppose we have a contact manifold (M,$\xi$) and two associated contact forms $\alpha$ and $\beta$ s.t.$\beta=f\alpha$ with $f>0$. Suppose also that we have two almost complex structures $J_\alpha$ ...

**4**

votes

**1**answer

408 views

### Recognizing the stablizer of a degenerate three forms in six dimension

Define $Stab^{+}(\Omega )$={ $\phi \in GL^{+}(V)$ : $\phi^{*}\Omega=\Omega$ }.
we say three-form $\Omega\in\wedge^{3}V^{*}$ is non-degenerate , if $i_X\Omega\neq 0$ for all $X\in V$-{0}
Let $V\cong ...

**1**

vote

**1**answer

177 views

### Enumerativity of Gromov-Witten invariants of orbifolds

For smooth Deligne-Mumford stacks, there is a well-defined Gromov-Witten theory, see http://arxiv.org/pdf/math/0103156.pdf and http://arxiv.org/pdf/math/0603151.pdf.
Is there some sense, or some ...

**1**

vote

**2**answers

261 views

### Proof of Arnold Conjecture for monotone symplectic manifolds

I have a question about the proof of the Arnold conjecture for monotone symplectic manifolds as it is explained in http://www.math.ethz.ch/~salamon/PREPRINTS/floer.pdf: Namely the author on page 32 ...

**1**

vote

**1**answer

149 views

### Condition in proof of the Arnold conjecture for monotone manifolds

In http://www.math.ethz.ch/~salamon/PREPRINTS/floer.pdf the Arnold conjecture from Symplectic Geometry is shown for the case of montone symplectic manifolds $(M, \omega)$ (i.e. we have $\int_{S^2} ...

**1**

vote

**1**answer

279 views

### What foliations are symplectic foliations?

On a manifold $M$, let $\mathcal F$ be a foliation having even-dimensional orientable leaves. I was wondering under what hypothesis I can state that $\mathcal F$ is the symplectic foliation of a ...

**10**

votes

**1**answer

846 views

### decompositions for exceptional Lie algebras $E_6$, $E_7$ and $E_8$

Do anybody know , why can we write the following decompositions for exceptional Lie algebras $E_6$, $E_7$ and $E_8$
1) $E_8=(V^{\star}\otimes \wedge^{8}V^{\star})\bigoplus ...

**5**

votes

**2**answers

374 views

### What information is required for SYZ mirror symmetry?

Let $X$ be a Calabi-Yau threefold. The Strominger-Yau-Zaslow conjecture suggests that $X$ should have a special Lagrangian $T^3$-fibration (when $X$ lies near a large complex structure limit) and a ...

**2**

votes

**1**answer

419 views

### why the group $GL(6,V)$ has an open orbit?

N.Hitchen in his paper about geometry of three forms wrote that "for a Real vector space $V$ of dimension six, the group $GL(6,V)$ has an open orbit and he referenced it to a thesis which was written ...

**8**

votes

**1**answer

478 views

### Known size invariant for Riemannian manifolds?

Larry Guth in his 2010 ICM address mentions the notion of a size invariant of Riemannian
metrics on a smooth manifold $M$. These are functions $S: Metrics(M) \to \mathbb{R}$ that are invariant under ...

**3**

votes

**2**answers

290 views

### Uniqueness of Kähler form with same volume

Hallo,
Let $M$ be a compact real-analytic Riemannian manifold with Riemannian metric $g$. Let $U \subset T^{*}M$ be a open neighbourhood of the zero section. On $U$ there exists a complex structure ...

**7**

votes

**2**answers

489 views

### Orientations for pseudoholomorphic curves with totally real boundary condition

I am trying to understand what the obstructions are to orienting moduli spaces of pseudoholomorphic curves with totally real boundary condition.
I believe that Fukaya-Oh-Ohta-Ono have shown that if ...

**5**

votes

**0**answers

260 views

### Sophus Lie on the symplectic foliation theorem

Given a Poisson manifold $(P,\{\cdot,\cdot\})$, its characteristic distribution $\mathcal C$ is the singular tangent distribution on $M$ generated by the Hamiltonian vector fields,i.e.
$$\mathcal ...

**5**

votes

**1**answer

279 views

### Questions on how SYZ conjectures is deduced from HMS conjeture.

The Strominge-Yau-Zaslow conjecture is roughly the following. Any Calabi-Yau $m$-manifold $X$ admits a special Lagrangian $T^m$ fibration (maybe at around a special point in its complex moduli space) ...

**4**

votes

**1**answer

214 views

### Smooth projective toric varieties which are quotients of product of spheres and torii by a free torus action?

My question is just as in the box. Is every smooth projective toric variety diffeomorphic to a quotient of $\prod_i S^{n_i} \times T^k$ (I know torus is a one-sphere but I just wanted to make clear I ...

**0**

votes

**1**answer

240 views

### $q_{S^*\omega}(X)=S^{\ast}q_{\omega}(X)$ ?

Definition: Let $(V,\Omega)$ be a symplectic vector space, we define
$\perp:\Lambda ^k(V^*)\to\Lambda ^{k-2}(V^{\ast})$
by $\perp(\omega)=i_{X_{\Omega}}(\omega)$
here if ...

**0**

votes

**0**answers

72 views

### Preimage of regular values of a admissible hamiltonian

Hi everyone: I have a question about a proof in Siburg's paper "Symplectic Capacities in Two Dimensions". We define a admissible Hamiltonian $H$ in such a way that near from the boundary of a ...

**2**

votes

**1**answer

195 views

### Hamiltonian actions and contractible loops

Let $(M, \omega)$ be a symplectic manifold and $G$ be a compact Lie group. Suppose we have a Hamiltonian $G$-action on $M$, with moment map $\mu: M \to {\mathfrak g}^*$.
We assume that the moment map ...

**4**

votes

**1**answer

200 views

### Rotations, Harmonic Oscillators, Gaussians, Ladders

I am trying to understand better the quantization of the Harmonic Oscillator.
Here are three ways of thinking about the Harmonic Oscillator.
Eigenfunctions of the differential operator: $H = ...

**4**

votes

**2**answers

628 views

### Applications of Floer homology

Can somebody tell me of other applications of Floer homology besides the proof of the Arnold conjecture.
Every answer would be appreciated.

**7**

votes

**3**answers

846 views

### Reasons for the Arnold conjecture

I am trying to understand the Arnold conjecture in Symplectic Geometry, which basically tells us the following: If $M$ is a compact symplectic manifold and $H_t$ be a 1-periodic Hamiltonian function, ...

**3**

votes

**1**answer

264 views

### Igusa invariants of genus 2 curves as Siegel modular functions?

Hi,
Are the Igusa invariants (defined in Igusa's oringal paper) also classical Siegel modular forms? I read from somewhere that
$\psi_4=\frac{1}{4}I_4, \quad \psi_6=\frac{1}{8}(I_2I_4-3I_6), \quad ...

**12**

votes

**1**answer

255 views

### Energy quantization for $J$-holomorphic spheres

Let $(\mathbb{CP}^1, j, g_{\text{FS}})$ be the complex projective line with the standard complex structure and the Fubini-Study metric and let $(M,J,\omega,g)$ be an almost Kähler compact manifold ...

**4**

votes

**1**answer

624 views

### an extended question of Gromov: Every **generalized open almost complex manifold** admits a **generalized symplectic structure**? [closed]

Definition (Open Manifolds):An open manifold is a manifold without boundary with no compact component. For a connected manifold, "open" is equivalent to "without boundary and non-compact.
we know that ...

**7**

votes

**0**answers

201 views

### From convex geometry to contact topology

Here is a problem in contact topology that was suggested by Petya's answer to this mathoverflow question of mine.
Let $S^* \mathbb{R}^n$ be the space of cooriented contact elements of $\mathbb{R}^n$. ...

**6**

votes

**1**answer

195 views

### Hamiltonian polar action with Lagrangian section

I am looking for examples of Hamiltonian polar isometric actions of a compact Lie group on a Kahler-Einstein (or perhaps just Kahler) manifold, that admits a Lagrangian section.
Recall that an ...