Some uniqueness problems in $\mathbb{H}^2\times\mathbb{R}$

In this talk we will consider two uniqueness problems in $\mathbb{H}^2\times \mathbb{R}$. First, we will prove a halfspace theorem for an ideal Scherk graph $S$ over a polygonal domain $D$ in $\mathbb{H}^2$, that is, we will show that a properly immersed minimal surface contained in $D\times \mathbb{R}$ and disjoint from $S$ is a translate of $S$. Second, we will consider a multi-valued Rado theorem for small perturbations of the Helicoid. More precisely, we will prove that for certain small perturbations of the boundary of a (compact) helicoid there exists only one minimal disk with that boundary. [mathjax]