Smoothness from elliptic theory I consider a smooth surface $N$ where the mean curvature $H$ is locally bounded and $N$ has locally uniform $C^1$ estimates. Then, a text states that 
" from elliptic theory, $N$ is smooth with estimates that are locally uniform."
Where does it come from ?

From the regularity theory for quasilinear elliptic partial differential equations of second order. See Gilbarg, Trudinger, Partial Differential Elliptic Equatinons second order for the general theory. I don't know whether that covers the result you are citing in full strength, but it will get you started. This is well known but non trivial (a lot of machinery) you need ot know. 
