Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 16 / 3

Fix a primitive $\lambda$ of the standard symplectic form on ${\mathbb{R}}^{2n}$. With the coordinates of Question 2 one convenient choice isFor a Lagrangian submanifold with inclusion $i:L\hookrightarrow {\mathbb{R}}^{2n}$, $d(i^{\ast }\lambda )=i^{\ast }{\omega }_0=0$. Its Liouville class of a Lagrangian submanifold isAny two primitives differ by a closed one-form on the contractible ambient space, hence by an exact form, so the class is independent of the primitive. The cotangent primitive convention $\theta ={\sum }_j{p}_{j}d{q}_j$, with ${\omega }_0=-d\theta$, gives the negative of this class; rationality, positive period size and nonvanishing do not depend on that sign.

The period homomorphism sends an integral one-cycle $c$ to ${\int }_c{i}^{\ast }\lambda$. Its image $\mathrm{Per}(L)\subset \mathbb{R}$ is an additive subgroup. A rational Lagrangian submanifold is one for which this subgroup is discrete. In the nonzero case it has a unique positive generator:Equivalently ${\ell }_L$ is a real multiple of an integral cohomology class. The invariant $\gamma (L)$ is its least positive Liouville period, also expressible as $\inf \{a>0:a\in \mathrm{Per}(L)\}$. If the period group is zero, that infimum is $+\infty$ by the empty-set convention; some treatments instead reserve “rational” for the nonzero discrete case. Because the ambient space is contractible, the Stokes theorem identifies these periods with signed symplectic areas of disks whose boundaries lie on $L$.

Let $\varphi \in {\mathrm{Ham}}_c({\mathbb{R}}^{2n})$ be nonidentity and choose $x$ with $\varphi (x)\ne x$. Choose disjoint open neighbourhoods $U,V$ of $x,\varphi (x)$, respectively. Continuity gives a small Euclidean ball $B=B(x,r_0)$ with $B\subset U$ and $\varphi (B)\subset V$. Therefore $\varphi (B)\cap B=\varnothing$.

Inside this ball construct an explicit closed rational Lagrangian submanifold. For $0<a<r_0/\sqrt{n}$, take the product Lagrangian torusIts distance from $x$ is $\sqrt{n}a<r_0$. Each coordinate circle is one-dimensional, so the pullback of $d{q}_j\wedge d{p}_j$ to it is zero; summing proves that this $n$-dimensional product is a Lagrangian submanifold. It is compact without boundary.

On the $j$th fundamental circle, $q_j=a\cos t$, $p_j=a\sin t$, and the primitive in part (a) restricts to $\frac{1}{2}{a}^{2}dt$. Its integral is $\pi a^2$. Translation of the torus adds only an exact constant-coefficient one-form to the primitive, so it does not change the periods. Since $H_1(T^n;\mathbb{Z})$ is generated by these circles,The given displacement-energy bound applies to this explicitly constructed torus. Because $L\subset B$, the map $\varphi$ displaces $L$, and henceThis is Hofer nondegeneracy from rational Lagrangian displacement. For two different maps $\varphi ,\psi$, apply it to ${\varphi }^{-1}\psi \ne \mathrm{id}$ and use the distance definition. Therefore ${\rho }_{\infty }(\varphi ,\psi )>0$. The equivalent oscillation Hofer metric is nondegenerate as well.

Only the stated energy inequality was used; its conclusion was not used to presume positive energy of an arbitrary open set. The rational torus and its positive period have been constructed inside the particular displaced ball.