be a fundamental pair on a surface ${\Sigma}$ with associated shape operator $S$, mean curvature $H$ and extrinsic curvature $K$. Let us suppose that every point $p\in\partial\Sigma_U$ has a neighborhood $V_p$ such that $$\frac{{\cal T}_{\widetilde{S}}}{H^2-K}\quad \mbox{is bounded in}\,
V_p\cap({\Sigma}-{\Sigma}_U),$$ where $\widetilde{S}=S-H\, Id$. Then either the Hopf differential of $(I,II)$ vanishes identically or its zeroes are isolated and of negative index.

In particular, if ${\Sigma}$ is a topological sphere then the pair is totally umbilical.

If $p\in\partial\Sigma_U$, then there exists an open neighborhood $V_p$ and a constant $m_0$ such that $\frac{{\cal T}_{\widetilde{S}}}{H^2-K}\leq m_0$ in $V_p\cap({\Sigma}-{\Sigma}_U)$. Thus, using Lemma \[anterior\] $$\label{noseyo}
|Q_{{\bar{z}}}|^2\ \leq\ m_0\,\frac{\lambda}{2}\,|Q|^2$$ in $V_p\cap({\Sigma}-{\Sigma}_U)$. Besides, since this inequality is also valid in the interior of $\Sigma_U$, we conclude that (\[noseyo\]) holds in $V_p$.

Therefore, using again [@AdCT] or [@Jost], we have that $p$ is an isolated zero of negative index of the Hopf differential, as we wanted to prove.

To end up, if ${\Sigma}$ is a topological sphere the result follows from the Poincaré index Theorem as in Theorem \[elteorema\].

Again, we point out that if ${\Sigma}$ is a topological torus under the assumptions above, then the pair is either totally umbilical or umbilically free. Analogously, if ${\Sigma}$ is a 