## Math 3600: Topics in Pure Mathematics : Symplectic Geometry

Tue, Th 4:00 - 5:15pm -- Thackeray 524

### Homework

• Homework 1 (due January 28)
Consider the ODE: $x'=b(x)$, where $b:\mathbb{R}^k\to\mathbb{R}^k$ is a sufficiently smooth function. Denote the flow of this ODE by $\phi_t(x)$.
1. Prove that $\big(\nabla\phi_t(x)\big)b(x) = b(\phi_t(x))$.
2. Prove that $\partial_t \big(\det\nabla\phi_t(x)\big) = \big((\mbox{div} b) (\phi_t(x))\big) \big(\det\nabla\phi_t(x)\big)$
3. Let $f:\mathbb{R}^k\to\mathbb{R}$ be a (smooth) scalar function on the phase space $\mathbb{R}^k$ and define: $v(x,t) = f(\phi_t(x))$. Prove that $v_t = \langle \nabla v, b\rangle$. This is called the Liouville theorem.

• Homework 2 (due February 4)
1. Prove the dual formulation of the Courant-Hilbert formula, which reads as follows. Denote the eigenvalues of a symmetric matrix $B\in\mathcal{S}(k)$ by: $\mu_1\leq\mu_2\ldots\leq\mu_k$. Then: $\mu_i = \sup_{\mbox{dim } V = j-1} \inf_{x\in V^\perp\setminus\{0\}}\frac{\langle Bx, x\rangle}{|x|^2}$.
2. Prove that eigenvalues of a Hamiltonian matrix $C\in Ham (n)$ where $C=JB$ and $B>0$ is a symmetric matrix, are purely imaginary and that they come in couples, namely: $\mbox{Spec}(C) = \{\pm i\lambda_j\}_{j=1}^n$ for some $\lambda_1\leq\lambda_2\ldots\leq\lambda_n$.
3. Let $T\in Sp(n)$ and write its unique polar decomposition: $T=PQ$ where $P\in S(2n)$, $P>0$ and $Q\in \mathcal{O}(2n)$. Prove that both $P$ and $Q$ are symplectic: $P, Q\in Sp(n)$.
4. Let $Q\in Sp(n)\cap\mathcal{O}(2n)$ and write $Q$ in the block form: $Q=\left[\begin{array}{cc} A & B \\ C & D\end{array} \right]$. Prove that $A=D$ and $B=-C$. Prove that $A-iB\in\mathbb{C}^{n\times n}$ is unitary.
5. Prove that $Q$ as in problem 4 above, is similar to $\mbox{diag}\big((A+iB), (A-iB)\big)$. Deduce that $\det Q>0$. Then, using problem 3 above deduce that for any $T\in Sp(n)$ there must be: $\det T = 1$. Note that this is an alternative proof of this statement, which does not use differential forms.

• Homework 3 (due February 16)
1. Show that the characteristic polynomial $P$ of a symplectic matrix is reflexive: $P(\lambda) = \lambda^{2n}P(\lambda^{-1})$. Deduce that the eigenvalues of a symplectic matrix occur in quadruples, namely: ($\lambda, \lambda^{-1}, \bar\lambda, \bar\lambda^{-1}$).
2. Let $H$ be a positive definite quadratic function on $\mathbb{R}^{2n}$. Show that there exists an antisymplectic linear map $T$ such that for every $x= (q_1\ldots q_n, p_1\ldots p_n)$ there holds: $H(Tx) = \sum_{j=1}^n \frac{q_j^2 + p_j^2}{r_j^2}$. Here, $0 < r_1\leq r_2\ldots \leq r_n$ are the symplectic radii of $H$.
3. Let $E_1$ and $E_2$ be two ellipsoids in $\mathbb{R}^{2n}$. Show that they have the same (vectors of) symplectic radii if and only if there is an antisymplectic linear map $T$ such that $T(E_1) = E_2$.
4. Compute the determinant of an antysymplectic linear transformation in $\mathbb{R}^{2n}$. What are the antisymplectic matrices in $\mathbb{R}^{2\times 2}$?
5. Let $T$ be an invertible linear map on $\mathbb{R}^{2n}$ for $n>1$. Assume that for every ellipsoid $E$, the second symplectic radii of $E$ and $TE$ are the same. Show that $T$ must be symplectic or antisymplectic.

• Homework 4 (due March 1)
1. Let $T$ be an invertibe $n\times n$ matrix. Show that the following conditions are equivalent:
(i) $T\in \mathcal{O}(n)$.
(ii) For every ellipsoid $E$ we have: $\mathbf{R}(E) = \mathbf{R}(TE)$, where $\mathbf{R}$ stands for the vector of the elliptic radii of a given ellipsoid.
(iii) Fix $i:1\ldots n$. For every ellipsoid $E$ we have: $\mathbf{R}_i(E) = \mathbf{R}_i(TE)$. Recall that $\mathbf{R}_i$ stands for the $i$-th elliptic radius (in order of magnitude) of a given ellipsoid.
2. Let $\varphi: \mathbb{R}^{2n}\to\mathbb{R}^{2n}$ be a symplectic diffeomorphism: $\phi\in Sp(\mathbb{R}^{2n})$. Let $H:\mathbb{R}^{2n}\to\mathbb{R}$ be a Hamiltonian. Prove that: $\phi_t^{(H\circ\varphi)} = (\varphi^{-1})\circ \phi_t^H\circ \varphi$. (Do not use the Cartan's formula!)
3. Prove that the Lie derivative of differential forms, defined as: $\mathcal{L}_X = d\circ i_X + i_X\circ d$ satisfies the following fundamental properties:
(i) $\mathcal{L}_X (\omega\wedge \eta) = \mathcal{L}_X (\omega)\wedge \eta + \omega\wedge \mathcal{L}_X (\eta)$,
(ii) $\mathcal{L}_X (d\omega) = d (\mathcal{L}_X \omega)$.
4. Let $(M_1, \omega^1)$ and $(M_2, \omega^2)$ be two symplectic manifolds. Prove that $(M_1\times M_2, \omega^1\times\omega^2)$ is a symplectic manifold, where we define: $(\omega^1 \times \omega^2)_{(x_1, x_2)} ((a_1, a_2), (b_1, b_2)) = \omega^1_{x_1}(a_1, b_1) + \omega^2_{x_2}(a_2, b_2)$ for all $(x_1, x_2)\in M_1\times M_2$ and all $a_1, b_i \in T_{x_1}M_1$, $a_2, b_2\in T_{x_2}M_2$.
5. Let $N$ be a $n$-dimensional smooth manifold. Define $\lambda\in \Lambda^1(T^*N)$ by: $\lambda_{(q,p)}(v,a) = p(v)$ for all $(q,p)\in T^*N$ and $(v,a)\in T_{(q,p)}(T^*N)$. Prove that $\omega=d\lambda$ is nondegenerate. This is done by defining appropriate (natural) charts $\bar h: T^*U\to\mathbb{R}^{2n}$ on each open cover set $U$ of $N$ such that: $(\bar h)^*\bar\lambda = \lambda$, where $\bar\lambda\in \Lambda^1(\mathbb{R}^{2n})$ is the usual: $\lambda = \langle p, dq\rangle$.

• Homework 5 (due March 15)
1. Let $(M, \omega)$ be a $2n$ dimensional, closed (i.e. compact, without boundary), symplectic manifold. Prove that for every $1\leq j \leq n$ there exists a closed $2j$ form which is not exact.
2. Let $(M, \omega)$ be a symplectic manifold of dimension higher than $2$. Let $\varphi:M\to M$ be a diffeomorphism and let $f:M\to\mathbb{R}$ be a Hamiltonian. Prove that if $\varphi^*\omega = f\omega$, then $f$ is constant.
3. Consider $M=\mathbb{R}^{2n-1}$ with $\alpha=\langle u, dx\rangle\in \Lambda^1(M)$ given by some smooth vector field $u$ on $M$.
(i) When is $(M,\alpha)$ contact?
(ii) Assume that $(M,\alpha)$ contact and let $H:\mathbb{R}^{2n}\to\mathbb{R}$ be a Hamiltonian function. Prove that the $\alpha$-contact field associated with $H$, i.e.: $Z=\pi_{T_xM} X_{\hat M}^{d\hat{\alpha}}$ is given by the formula: $$Z= -\tilde C \nabla H + HR^\alpha,$$ where $R^\alpha$ is the associated Reeb vector field and $\tilde C\in\mathbb{R}^{(2n-1)\times (2n-1)}$ is uniquely specified by the two requirements: $(\tilde C)_{\mid u^\perp}$ is the inverse of the linear map $(\nabla u) - (\nabla u)^T: u^\perp \to (R^\alpha)^\perp$; and $\tilde C u = 0$.
(iii) Assume that $n=2$. Find what is $\tilde C$ in this case and prove that there must be: $Z = {\langle u, \mbox{curl} u\rangle}^{-1}\big(H\mbox{curl} u - u\times \nabla H\big).$
4. Let $M$ be an odd-dimensional manifold and let $\alpha\in \Lambda^1(M)$. Prove that $\alpha$ is contact if and only if $\alpha\wedge (d\alpha)^{n-1}$ is a volume form on $M$.
5. Let $(M,\omega)$ be a symplectic manifold of dimension larger than $2$. Let $\phi:M\to M$ be a diffeomorphism such that $\phi^*\omega = f\omega$ for some scalar function $f:M\to\mathbb{R}$. Prove that $f$ must be constant.

• Homework 6 (due March 31)
1. Let $M=\mathbb{T}^3$ be the $3$-dimensional torus and define $\hat M = M\times (-\delta, \delta)$ for some $\delta>0$. For a vector $(\xi_1, \xi_2, \xi_3)\in\mathbb{R}^3$ such that $\xi_3\neq 0$, consider the skew-symmetric matrix: $$\bar C = \left[\begin{array}{cccc} 0 & 1 & 0 & \xi_1\\ -1 & 0 & 0 & \xi_2 \\ 0 & 0 & 0 & \xi_3 \\ -\xi_1 & -\xi_2 & -\xi_3 & 0\end{array}\right].$$ Let $C= - \bar{C}^{-1}$ and define: $\omega = \sum_{i < j} c_{ij} dx_i\wedge dx_j\in\Lambda^2(\hat M)$. Finally, define the Hamilonian on $\hat M$ by: ${H}(x_1, x_2, x_3, x_4)=x_4$.
(i) Prove that $(\hat M, \omega)$ is a symplectic manifold.
(ii) Compute the Hamiltonian vector field $X_H^\omega$.
(iii) Assume that $\xi$ is irrational, that is for all $a\in \mathbb{Z}^3\setminus \{0\}$ we have: $\langle \xi, a\rangle \neq 0$. Prove that $H$ has then no periodic orbit. Deduce that the submanifold $M$ has no contact structure in $(\hat M, \omega)$.
2. Let $M=\partial D$ for an open, bounded domain $D\subset\mathbb{R}^{2n}$ that is star-shaped with respect to a ball $B_\epsilon(0)$. Prove that $M$ is a contact manifold. [Hint: Check that $X(x) = \frac{1}{2}x$ is $\omega$-Liouville.]
3. Consider the contact manifold $(\mathbb{R}^{2n-1}, \bar\alpha)$, where we write $x=(q,p,z)$ and $\bar\alpha = \langle p, dq\rangle + dz$. Let $H=H(q,p,z)$ be a smooth Hamiltonian on $\mathbb{R}^{n-1}$. Show that the related contact vector field $Z_H$ is given by: $Z_H = \big (\partial_pH , -\partial_qH + \partial_zH p , H - \langle p, \partial_pH\rangle\big)$.
4. Consider the manifold $M=\mathbb{S}^{2n-1}\subset\mathbb{R}^{2n}$. Let $\bar\mu\in \Lambda^1(M)$ be the restriction of $\bar\lambda=\langle p, dq\rangle\in \Lambda^1(\mathbb{R}^{2n})$, where we wrire $x=(q,p)\in\mathbb{R}^{2n}$.
(i) Show that $(M, \bar\mu)$ is not a contact manifold.
(ii) Let $\mu = \frac{1}{2} \big(\langle p, dq\rangle - \langle q, dp\rangle\big)\in \Lambda^1(M)$. Show that $(M, \mu)$ is a contact manifold. Find its Reeb vector field.
5. Let $K=\bar{U}$ for an open, convex set $U\subset\mathbb{R}^{2n}$ containing $0$. Recall that the gauge function of $K$ is defined as: $g_K(x) = \inf\{r>0; ~ \frac{x}{r}\in K\}$.
(i) Prove that $g_K$ is a Banach functional.
(ii) Compute the Legendre transform of $g_K$.
(iii) Prove that $K=\{y; ~ \langle x, y\rangle\leq\sup_{z\in K} \langle x, z\rangle \quad \forall x\in \mathbb{R}^{2n}\}$.

• Homework 7 (due April 12)
1. Consider the Hamiltonian $H:\mathbb{R}^2\to\mathbb{R}$ given by: $H(x) = \pi |x|^2$ and the periodic solutions of $\dot x=J\nabla H(x)$ on the level set $\{H=\frac{1}{2}\}$.
(i) Show that $H=H_K$ for some set $K$. What is $K$?
(ii) Show that $\inf_{\Lambda} \mathcal{C} = 0$.
(iii) Show that $\sup_{\Lambda} \mathcal{C} = \infty$.
2. Let $K\subset \mathbb{R}^n$ be the closure of an open set $U$, that is convex and symmetric with respect to $0$. Show that the "Minkowski Billiard" Hamiltonian flow starting from $(0, \nabla g_K(q))\in K\times K^o$ for $q\in\partial K$, induces a periodic orbit whose action equals $4$. Deduce that the symplectic width of the set $K\times K^o\subset\mathbb{R}^{2n}$ satisfies: $\mathbb{c}(K\times K^o) \leq 4$.
3. Prove that for every symplectic manifold $(M, \omega)$ there holds: $\underline{\mathbb{c}}(M, \omega) \leq \mathbb{c}(M, \omega)\leq \overline{\mathbb{c}}(M, \omega)$.
4. Show that $\mathbb{c}_{HZ}(B_{2n}(0,1), \bar\omega) \geq \pi$.
5. Show that $\mathbb{c}_{HZ}(K, \bar\omega) \geq \mathbb{c}_0(K)$ for every bounded convex set $K\subset\mathbb{R}^{2n}$ that is a closure of an open set.

• Homework 8 (due April 26)
1. Verify the symplectomorphism monotonicity and the scaling properties in the definition of symplectic capacity for $\mathbb{c}_{HZ}$.
2. Assume that $F\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R})$ satisfies $\lim_{|x|\to\infty}F(x) = +\infty$.
(i) Show that $F$ automatically satisfies the Palais-Smale condition.
(ii) Assume that the flow of $x'=-\nabla F(x)$ is well defined for all $t\geq 0$. Let $x_1, x_2$ be two distinct relative minima of $F$. Prove that there exists yet another critical point $x_3\not\in\{x_1, x_2\}$.
3. Consider the functon $F(x,y) = e^{-x} - y^2$.
(i) Show that it does not satisfy the Palais-Smale condition.
(ii) Denote $E^\pm = \{(x,y); ~ F(x,y)\leq 0, ~ \pm y \geq 0\}$, and let $\mathcal{F}$ be a family of subsets of $\mathbb{R}^2$ given by: $\mathcal{F}= \{\gamma[0,1]; ~ \gamma:[0,1]\to\mathbb{R}^2, ~ \gamma(0)\in E^-, ~\gamma(1)\in E^+, ~ \gamma \mbox{ continuous}\}.$ Show that $\mathcal{F}$ is positively invariant with respect to the flow of $(x,y)' = -\nabla F(x,y)$ and that $\alpha(F, \mathcal{F}) = 0$, but $F$ has no critical point.