On the Ekeland-Hofer symplectic capacities of the real bidisc

In $\mathbb{C}^2$ with the standard symplectic structure we consider the bidisc $D^2\times D^2$ constructed as the product of two open real discs of radius $1$. We compute explicit values for the first, second and third Ekeland-Hofer symplectic capacity of $D^2\times D^2$. We discuss some applications to questions of symplectic rigidity.Comment: v3: Final version, to appear in "Pacific J. Math.", 20 page