The gap between near commutativity and almost commutativity in symplectic category

It is enough to construct a smooth embedding of $B^{2}(\pi)$ into $B^{2n}(2\pi)$ (where $B^{2n}(2\pi)$ is the $2n$-dimensional ball of capacity $2\pi$, or equivalently, radius $\sqrt{2}$, centred at the origin), such that its image is a symplectic curve having arbitrarily large symplectic area. An example of such an embedding is the map $u:B^{2}(\pi)\rightarrow\mathbb{C}^{n}$ given by $u(z)=(z^{k},z,0,0,...,0)$, where $k\in\mathbb{N}$ is large enough. ∎

The proof goes similarly as the proof of Proposition 2.2 (which is Proposition 1.21 in [BEP]). ∎

Consider a Darboux neighborhood $U\subset M$. Denote $l=1/\sqrt{q}$, $\epsilon=1/p-1/q$. Then $q=1/l^{2}$, $p=1/(l^{2}+\epsilon)$. By Lemma 2.3, there exists a symplectic embedding $u:(B^{2}(2l^{2}+3\epsilon),\sigma_{std})\rightarrow(U,\omega)$. By the symplectic neighborhood theorem, a neighborhood of $u(B^{2}(2l^{2}+3\epsilon))$ contains an open subset which is symplectomorphic to the product $B^{2}(2l^{2}+2\epsilon)\times B^{2}(\epsilon^{\prime})^{\times(n-1)}$ of a disc of area $2l^{2}+2\epsilon$ and $n-1$ copies of a disc of area $\epsilon^{\prime}$, for some $\epsilon^{\prime}>0$. Let $\phi:(W=B^{2}(2l^{2}+2\epsilon)\times B^{2}(\epsilon^{\prime})^{\times(n-1)},\omega_{std})\rightarrow(M,\omega)$ be a symplectic embedding whose image is this open subset. Choose a curvilinear quadrilateral inside $B^{2}(2l^{2}+2\epsilon)$ with sides $a_{1},a_{2},a_{3},a_{4}$ (which are given in cyclic order), and area $l^{2}+\epsilon$.

Consider functions $\tilde{f}_{1},\tilde{g}_{1}$, as in Claim 2.5. Choose $0<\rho_{1}<\rho<\rho_{2}$ such that $\pi\rho_{2}^{2}<\epsilon^{\prime}$, and choose a smooth function $\tilde{h}_{1}:B^{2}(\epsilon^{\prime})\rightarrow\mathbb{R}$ such that its support lies inside the annulus $D^{2}_{\rho_{1},\rho_{2}}=\{z\in B^{2}(\epsilon^{\prime})\,|\,\rho_{1}<|z|<\rho_{2}\}$, such that $0\leqslant\tilde{h}_{1}\leqslant 1$ on $B^{2}(\epsilon^{\prime})$, and such that $\tilde{h}_{1}=1$ on $S^{1}_{\rho}=\{z\in B^{2}(\epsilon^{\prime})\,|\,|z|=\rho\}$. Now define $f_{1},g_{1}:W\rightarrow\mathbb{R}$ by

for $(z,w_{1},...,w_{n-1})\in W$, and then define $f,g:M\rightarrow\mathbb{R}$ by setting $f=g=0$ on $M\setminus\phi(W)$ and setting $f(\phi(x))=f_{1}(x)$, $g(\phi(x))=g_{1}(x),$ for any $x\in W$. First note that $f_{1}=0$ on $a_{1}\times(S^{1}_{\rho})^{n-1}$, $f_{1}=1$ on $a_{3}\times(S^{1}_{\rho})^{n-1}$, $g_{1}=0$ on $a_{2}\times(S^{1}_{\rho})^{n-1}$, and $g_{1}=1$ on $a_{4}\times(S^{1}_{\rho})^{n-1}$. Secondly, we have

for any $(z,w_{1},...,w_{n-1})\in W$, and hence $\|\{f,g\}\|=q$. Also we have that $f,g\geqslant 0$ on $M$, and that $\|f\|=\|g\|=1$. Let us show that the constructed functions $f,g$ satisfy (1.1). For showing the upper bound, choose a smooth compactly supported function $h:M\rightarrow[0,1]$ such that $h=1$ on the union of supports $\text{supp}(f)\cup\text{supp}(g)$, and define functions $F,G:M\rightarrow\mathbb{R}$ by $F=(\frac{1}{2}-\frac{s}{2q}+\frac{s}{q}f)h$, $G=g$ (the upper bound in (1.1) is essentially the statement of Theorem 1.4 (ii), inequality (8) in [BEP]). Let us show the lower bound, for a given value of $s$. Since the profile function $\rho_{f,g}$ is non-negative and non-increasing, it is sufficient to prove the lower bound for $s\in(0,p)$. Take $s\in(0,p)$ and denote $t=\frac{1}{2}-\frac{1}{2p}s>0$. Now assume that we have a pair of smooth compactly supported functions $F,G\in C^{\infty}_{c}(M)$, such that $\|F-f\|+\|G-g\|\leqslant t$. Denote $\alpha=\|F-f\|$, $\beta=\|G-g\|$. Then $\alpha,\beta\geqslant 0$ and $\alpha+\beta\leqslant t$. Now choose $\delta>0$ small, and pick smooth functions $u,v:\mathbb{R}\rightarrow\mathbb{R}$, such that the function $u$ satisfies $u(x)=0$ for $x\in[-\alpha,\alpha]$, $|u(x)-x|\leqslant(1+\delta)\alpha$ for $x\in\mathbb{R}$, and $|u^{\prime}(x)|\leqslant 1$ for $x\in\mathbb{R}$, and such that the function $v$ satisfies $v(x)=0$ for $x\in[-\beta,\beta]$, $|v(x)-x|\leqslant(1+\delta)\beta$ for $x\in\mathbb{R}$, and $|v^{\prime}(x)|\leqslant 1$ for $x\in\mathbb{R}$. Then the functions $F^{\prime}=u\circ F$, $G^{\prime}=v\circ G$ satisfy $\|\{F^{\prime},G^{\prime}\}\|\leqslant\|\{F,G\}\|$. But moreover, we have that $F^{\prime}(x)=0$ whenever $f(x)=0$, $F^{\prime}(x)\geqslant 1-(2+\delta)\alpha$ whenever $f(x)=1$, $G^{\prime}(x)=0$ whenever $g(x)=0$, and $G^{\prime}(x)\geqslant 1-(2+\delta)\beta$ whenever $g(x)=1$, for any $x\in M$. Hence if we denote $W^{\prime}=B^{2}(2l^{2}+2\epsilon)\times(D^{2}_{\rho_{1},\rho_{2}})^{\times(n-1)}\subset W$, then the supports of $F^{\prime},G^{\prime}$ lie inside $\phi(W^{\prime})$, so in particular the supports of $\phi^{*}F^{\prime},\phi^{*}G^{\prime}$ lie inside $W^{\prime}$, and moreover we have that $\phi^{*}F^{\prime}=0$ on $a_{1}\times(S^{1}_{\rho})^{n-1}$, $\phi^{*}F^{\prime}\geqslant 1-(2+\delta)\alpha$ on $a_{3}\times(S^{1}_{\rho})^{n-1}$, $\phi^{*}G^{\prime}=0$ on $a_{2}\times(S^{1}_{\rho})^{n-1}$, and $\phi^{*}G^{\prime}\geqslant 1-(2+\delta)\beta$ on $a_{4}\times(S^{1}_{\rho})^{n-1}$. Consider a symplectic embedding $\psi$ of $W^{\prime}=B^{2}(2l^{2}+2\epsilon)\times(D^{2}_{\rho_{1},\rho_{2}})^{\times(n-1)}$ into $\mathbb{R}^{2}\times(T^{*}S^{1})^{\times(n-1)}=\mathbb{R}^{2}\times T^{*}\mathbb{T}^{n-1}$ (where $\mathbb{T}^{n-1}=(S^{1})^{n-1}$ is the $(n-1)$-dimensional torus), given as a product of maps, such that at the factor $B^{2}(2l^{2}+2\epsilon)$ we have the standard embedding into $\mathbb{R}^{2}$ (given by the identity map), and such that at each factor $D^{2}_{\rho_{1},\rho_{2}}$, the circle $S^{1}_{\rho}$ is mapped onto the zero section of $T^{*}S^{1}$. Denote the push-forwards $F_{2}=\psi_{*}\phi^{*}F^{\prime}=F^{\prime}\circ\phi\circ\psi^{-1}$, $G_{2}=\psi_{*}\phi^{*}G^{\prime}=G^{\prime}\circ\phi\circ\psi^{-1}$, which are a priori defined on $\psi(W^{\prime})$, and extend them by $0$ to obtain functions on the whole $\mathbb{R}^{2}\times T^{*}\mathbb{T}^{n-1}$. We have $F_{2}=0$ on $a_{1}\times L_{0}$, $F_{2}\geqslant 1-(2+\delta)\alpha$ on $a_{3}\times L_{0}$, $G_{2}=0$ on $a_{2}\times L_{0}$, and $G_{2}\geqslant 1-(2+\delta)\beta$ on $a_{4}\times L_{0}$, where $L_{0}\subset T^{*}\mathbb{T}^{n-1}$ is the zero section. Therefore in view of Lemma 2.4, we get

Since we can choose $\delta>0$ to be arbitrarily small, we in fact get

Thus we have shown that for any $F,G\in C^{\infty}_{c}(M)$ with $\|F-f\|+\|G-g\|\leqslant t$ we have $\|\{F,G\}\|\geqslant s$. This immediately implies

∎