# Questions tagged [sg.symplectic-geometry]

Hamiltonian systems, symplectic flows, classical integrable systems

1,084
questions

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### How to check conditions for Liouville-Arnold theorem?

Arnold gives in his book "Mathematical Methods of Classical Mechanics" on p.272 the following, well known theorem:
Let $F_1, \dots, F_n$ be $n$ functions in involution on a symplectic $2n$-...

**6**

votes

**2**answers

291 views

### Symplectisation as a functor between appropriate categories

Let $(M,\xi)$ be a transversally orientable contact manifold, that is, there exists a form $\alpha \in \Omega^1(M)$ such that $\xi = \ker \alpha$. Then we can associate to $(M,\xi)$ its ...

**5**

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**1**answer

243 views

### Diffeomorphic but not isotopic symplectic forms

Do we know of any closed symplectic manifold $M$ with 2 cohomologous symplectic forms $\omega_1$ and $\omega_2$ such that there exist $\psi \in \text{Diff}(M)$ and $\psi^* \omega_1 = \omega_2$ but $\...

**3**

votes

**1**answer

139 views

### Mirror symmetry for singular Lagrangian torus fibrations

Let $X$ be a closed symplectic manifold equipped with a smooth Lagrangian torus fibration $\pi:X \rightarrow Q$. Assume that $\pi$ admits a Lagrangian section. By work of Kontsevich-Soibelman, one can ...

**4**

votes

**2**answers

231 views

### Are symplectomorphisms of Weil–Petersson symplectic form induced from surface diffeomorphisms?

Let $S$ be a closed hyperbolic surface of genus $g\geq 2$. Let $(\mathcal{T},\omega)$ be the corresponding Teichmuller space with the Weil–Petersson symplectic from $\omega$. Let $\Phi:\mathcal{T}\...

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### Example of overtwisted contact manifold with finitely many periodic Reeb orbits

Are there examples of overtwisted manifolds with only a finite number of periodic Reeb orbits?
An example is given by the irrational ellipsoid in $(\mathbb{R}^4,\omega_\text{st})$, which is not ...

**4**

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**1**answer

163 views

### Moment map interpretation of Einstein equation

Einstein's famous equation relates the geometry of a (4-dimensional) manifold to the matter content in that manifold.
Is there a way to obtain Einstein's equation as a moment map?
More precisely, ...

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### Star-shaped domain in $\mathbb{C}P^2$

Consider $(\mathbb{C}P^2,\omega_{FS})$ where $\omega_{FS}$ is the standard Fubini-Study form. Let $L$ denote a sphere in $\mathbb{C}P^2$ in the class $\mathbb{C}P^1$. Further let $\int_{L} \omega_{FS} ...

**3**

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**1**answer

166 views

### One-Forms in Functional Space?

I am presently reading this paper on covariant phase space and I have difficulty understanding the following formalism developed:
In the paper (section $2.2$, pg. $12$), the authors have introduced ...

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94 views

### A non-Kaehler manifold complex and symplectic in exactly one way

Does there exist a closed connected smooth manifold that admits exactly one (up to biholomorphism) integrable complex structure and exactly one (up to symplectomorphism and rescaling) symplectic ...

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**1**answer

112 views

### Lagrangian torus fibrations and Arnol'd-Liouville theorem

Let $(X, \omega)$ be a closed symplectic manifold of dimension $2n$ and $\pi: X \rightarrow Q$ a Lagrangian torus fibration. Let $F_q$ denote the fiber at $q \in Q$. It is claimed in a paper of ...

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**1**answer

135 views

### Log Calabi-Yau surfaces without maximal boundaries

Let $X$ be a smooth projective surface over $\mathbb{C}$, $D\subset X$ is an effective divisor. $(X,D)$ is a log Calabi-Yau pair if $K_X+D$ is a principal divisor. The complement $M=X\setminus D$ is a ...

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104 views

### Orientable surface bundle

Is it true that every orientable surface bundle can be made into a symplectic fibration?If yes, why?
What about the particular case that $M$ is a connected compact 4-manifold?

**2**

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**1**answer

115 views

### Reference for the nearby Lagrangian conjecture for $T^*S^1$

I am looking for a reference for the proof of the nearby Lagrangian conjecture of $T^*S^1$, that is, that every exact and compact Lagrangian submanifold of the cylinder is Hamiltonianly isotopic to ...

**3**

votes

**1**answer

115 views

### Compactly supported symplectomorphisms of $D^2$

I'm trying to understand a more basic version about Gromov's theorem about the compactly supported symplectomorphisms of $D^2 \times D^2$ being contractible.
Consider the dimensional disk $D^2 \...