Symplectic Rigidity of Real Bidisc

Abstract

Let $\mathbb{D}$ be the unit disc in $\mathbb{C}$, then $\mathbb{D}^n(r)$ is the complex or symplectic $n$-discs of radius $r$. Let $z_j = x_j+iy_j\in\mathbb{C}, j=1,2$ and $\mathbb{D}_{\mathbb{R}}^2=\{(z_1,z_2) : |x_1|^2+|x_2|^2<1,|y_1|^2+|y_2|^2<1\}$ be the real bidisc. In this paper we will prove the following two theorems: 1) If $T\in O(4)$ is an orthogonal transformation on $\mathbb{R}^4$, then $T(\mathbb{D}^2)$ is symplectomorphic to $\mathbb{D}^2$ w.r.t. the standard symplectic form on $\mathbb{R}^4$ if and only if $T$ is unitary or conjugate to unitary. 2) For $r\geq 1$ and $n\geq 2$, $\mathbb{D}_{\mathbb{R}}^2\times \mathbb{D}^{n-2}(r)$ and $\mathbb{D}^2\times \mathbb{D}^{n-2}(r)$ are not symplectomorphic w.r.t. the standard symplectic form on $\mathbb{C}^n$.