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I am a bit confused about Theorem 2.16 in the book "A Course in Minimal Surfaces" by Colding and Minicozzi. The authors write that Theorem 2.16 was proved in this 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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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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$A$ denotes the second fundamental form.

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  • $\begingroup$ Could you state Theorem 2.16 in your question? $\endgroup$ – Deane Yang Nov 20 '19 at 15:28
  • $\begingroup$ @DeaneYang Thanks for your comment. I just edited the question, including the statement of Thm 2.16 with a remark. $\endgroup$ – Math_tourist Nov 20 '19 at 16:47
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    $\begingroup$ Sorry. One more quibble. You should say what $A$ is. Looks like it's the second fundamental form. $\endgroup$ – Deane Yang Nov 20 '19 at 16:54
  • $\begingroup$ I believe a proof of the statement with assumption (ii) can be found in a paper by Schoen, Simon, and Yau titled Curvature estimates for minimal hypersurfaces. $\endgroup$ – Deane Yang Nov 20 '19 at 16:56
  • $\begingroup$ ok! Thanks a lot!! Do you know if one can get any pointwise curvature estimate for a general quasiconformal surface assuming condition (i)? (As I wrote above, I don't expect such a strong estimate as in the statement of THM 2.16, because of the counterexample given by the cylinder. But maybe something weaker? Like $|A| \in L^{\infty}(\Sigma)$? $\endgroup$ – Math_tourist Nov 20 '19 at 20:49
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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.

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