Abstract: Allard’s regularity theorem proves that an n-dimensional integral varifold whose mass ratio is close to 1 in a given ball and has generalized mean curvature in $L^p$ with p＞n is in fact a $C^{1,\alpha}$ graph at a slightly smaller scale. In $p=n$ case one cannot hope for a $C^{1,\alpha}$ result, as $W^{2,n}$ graphs easily show. However, it is long known in the literature (since the pioneering works of Toro and M\"{u}ller-\v{S}verak in the nineties) that, for a 2-dimensional surface, an $L^2$ control of the whole second fundamental form allows for bi-Lipschitz parametrization. Inspired by Toro and M\"{u}ller-\v{S}verak' work, we obtain an extension of Allard (when $p=n=2$) showing that (when the mass ratio is sufficiently small), the varifold is (at a slightly smaller scale) bi-Lipschitz homeomorphic to a disk.