Is the manifold $S^1\times S^1 \times D^2$ a hyperkaehler manifold?

Question

Let $D^2$ be a disk. Is it possible to show that $S^1\times S^1 \times D^2$, is hyperkaehler?

I have learnt from here that in order for this to be true, there should be an immersion of $S^1\times S^1 \times D^2$ in $\mathbb{R}^4$. However, I have not been able to show this.

Answer 1

$\mathbb S^1 \times \mathbb S^1 \times D^2$ is clearly hyperkahler: the standard flat metric $$ g = d\theta_1^2 + d\theta_2^2 + dx^2 + dy^2$$ is an hyperkahler matric.