I am a bit confused about Theorem 2.16 in the book ["A Course in Minimal Surfaces"](https://bookstore.ams.org/gsm-121) by Colding and Minicozzi. The authors write that Theorem 2.16 was proved in [this](https://www.jstor.org/stable/j.ctt1b7x7tv.9?refreqid=excelsior%3Ad83238340e13a211195956221927fefe&seq=1#metadata_info_tab_contents) paper by Schoen and Simon in the more general setting of surfaces with quasiconformal Gauss map. 


I looked at the paper of Schoen and Simon, but they prove a Holder estimate for the Gauss map. How can one get the curvature estimate in Colding-Minicozzi's book from that Holder estimate? 


Any help will be very much apreciated!


This is the statement of Theorem 2.16. 


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**EDIT**



> 
> **Theorem 2.16** : Let $0 \in \Sigma \subset B\_{r\_0} = B\_{r\_0}(0) \subseteq \mathbb{R}^3$ be an embedded simply connected minimal surface with $\partial \Sigma \subset \partial B\_{r\_0}$. If $\mu > 0$ and either 
>  \begin{equation}(i)\qquad Area(\Sigma) \le \mu r\_0^2 \end{equation}
>  or
>  \begin{equation} (ii) \qquad \int\_{\Sigma}|A|^2 \le \mu, \end{equation}
>  then for the connected component $\Sigma'$ of $B\_{\frac{r\_0}{2}} \cap \Sigma$ with $0 \in \Sigma'$ we have
>  \begin{equation}(\star) \qquad \sup\_{\Sigma'}|A|^2 \le C r\_0^{-2} \end{equation}
>  for some $C = C(\mu)$.
> 
> 
> 


**I know that one cannot expect such a strong estimate for a general surface with quasiconformal Gauss map.** For instance consider a cylinder $\mathbb{S}^1 \times \mathbb{R}$. It has quasiconformal Gauss map because it has bounded (actually constant) mean curvature. Moreover it has linear area growth and thus in particular it satisfies $(i)$ for large $r\_0$. But it does not satisfy $(\star)$. 


On the other hand from the paper of Schoen and Simon I would expect to derive a pointwise curvature estimate, but I don't know how.


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**SECOND EDIT**: 


$A$ denotes the second fundamental form.



Curvature estimate for minimal surfaces
Theorem 2.16 (for embedded minimal disks) is implicit in Schoen-Simon. There are two steps. The first is that some area (or total curvature) bound on an extrinsic ball implies small total curvature on a sub-ball. This is the key point and it is this that is generalized to intrinsic balls here.


Schoen and Simon show that the area bound implies small total curvature on a sub-ball even for surfaces with quasi-conformal Gauss maps. The argument that gives this also implies a Holder estimate on the Gauss map.


The second part, which is true for minimal surfaces but not necessarily more generally, is that small total curvature implies a point-wise estimate. There are many ways to see this. The simplest is via the Choi-Schoen estimate. One could also see it from Schoen-Simon-Yau in this case since one also has area bounds.


