closed topological disk and $\partial{\Sigma}$ is a line of curvature of $(I,II)$, then the pair is totally umbilical.

If $(I,II)$ is a Codazzi pair we have $$T_{\widetilde{S}}\left(\frac{\partial\ }{\partial z},\frac{\partial\ }{\partial
{\bar{z}}}\right)=-\nabla_{\frac{\partial\ }{\partial z}}H\frac{\partial\
}{\partial {\bar{z}}}+\nabla_{\frac{\partial\ }{\partial {\bar{z}}}}H\frac{\partial\
}{\partial z}=H_{{\bar{z}}}\frac{\partial\
}{\partial z}-H_{z}\frac{\partial\
}{\partial {\bar{z}}}$$ and, therefore, its Codazzi function is $${\cal T}_{\widetilde{S}}\,=\,\frac{2}{\lambda}\,|H_{z}|^2\,=\,\|\nabla H\|^2,$$ where $\|\nabla H\|$ stands for the modulus of the gradient of $H$ for the Riemannian metric $I$. Thus, Theorem \[thglover\] can be applied for Codazzi pairs whenever the quotient $\|\nabla
H\|^2/(H^2-K)$ is bounded.

However, this result can also be applied to fundamental pairs which are not Codazzi pairs, as was implicitly made in [@EGR] in order to classify the complete surfaces with constant extrinsic curvature in the product spaces $\h^2\times\r$ and $\s^2\times\r$.

Holomorphic quadratic differentials. {#s4}
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In this section we will see that, under certain assumptions on a Codazzi pair, it is possible to obtain a new Codazzi pair with vanishing constant mean curvature which is geometrically related to the first one. Thanks to this second Codazzi pair, we will show the existence of a holomorphic quadratic differential which will provide important information on the geometric behavior of the initial pair.

If $(u,v)$ are doubly orthogonal parameters 