I have two questions.

1. I consider a sequence of metrics $h_n$ on a two-dimensional torus which all induce the same conformal structure. Suppose that the volume of $h_n$ is always $1$. Is it possible that the diameter of $h_n$ tends to infinity?

2. Fix a conformal structure on a torus. Can I holomorphically embed cylinders of arbitrarily large modulus in this torus?

Thanks for your attention.

Selim