The Hoelder continuity in the interior is an outcome of Almgren's original theory, to which C. De Lellis and E.N. Spadaros work has given simpler alternative approaches. The Hoelder continuity in the interior is known and can be found in C. De Lellis and E.N. Spadaros work.   First I will introduce the concept of Almgren's Q-valuded functions and give a short overview of the known results. Thereafter I would like to point out why and how classical methods proving boundary regularity fail. Why the approach used in the interior cannot work. Towards the end I will present details to the proof.

In this talk I will present several results on the geometry of surfaces immersed in $\mathbb R^3$ with bounded $L^2$ norm of the norm of the second fundamental form $|A|$ and $L^p$, $p>2$ norm of the mean curvature $H$. In particular we will show how "smallness" on the bound of any of the two aforementioned quantities implies that the surface is graphical. This is joint work with G. Tinaglia.