An Allard type regularity theorem for varifolds with Hölder continuous generalized normal

We prove that Allard's regularity theorem holds for rectifiable \(n\)-dimensional varifolds \(V\) assuming a weaker condition on the first variation. This, in the special case when \(V\) is a smooth manifold translates to the following: If \(\omega_n^{-1}\rho^{-n}{\rm area}(V\cap B_\rho(x))\) is sufficiently close to \(1\) and the unit normal of \(V\) satisfies a \(C^{0,\alpha}\) estimate, then \(V\cap B_{\rho/2}(x)\) is the graph of a \(C^{1,\alpha}\) function with estimates. Furthermore, a similar boundary regularity theorem is true.