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Assume $M^n$ is a compact hypersurface without boundary immersed in $R^{n+1}$, with $A$ its 2nd fundamental form. If the square norm of A is bounded by an abstract constant, i.e. $|A|^2\leq C$ for some constant $C$. Question: Can one pick up a constant $r$ only depends on $C$ and $n$ s.t. for any point $p$ on $M^n$, $M^n$ can be written as a (local) graph in an $n+1$ ball $B_r(p)$. Thanks! Rmk. Notice that $r$ is independent of $M^n$!

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    $\begingroup$ Could you give some context and motivation? This looks like a high-level homework without them (hint: your assumption bounds the derivative of the Gauss map). $\endgroup$
    – Benoît Kloeckner
    Commented Nov 24, 2013 at 9:17

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This is a standard fact which is often asserted in the literature. A proof is given here: http://www.ugr.es/~jmmanzano/santalo/notes/GiuseppeTinaglia-Santalo-Granada.pdf (section 1), but I don't know of the "original" reference. I think there is also something in Colding--Minicozzi's book on minimal surfaces, but I don't have that in front of me right now.

As for some motivation, you may be interested in Ch 7 of these notes I took in a course by Brian White: http://math.stanford.edu/~ochodosh/MinSurfNotes.pdf, where this fact is combined with Arzelà--Ascoli to get compactness of sequences of surfaces with locally controlled second fundamental form. This is then used to prove curvature estimates for minimal (and other) surfaces by "blowup," which is a very beautiful set of ideas.

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Perhaps a paper of Prof. Ralph Howard could provide an answer. http://people.math.sc.edu/howard/Notes/schur.pdf

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