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 $ω_n^{-1}ρ^{-n}{\rm area}(V\cap B_ρ(x))$ is sufficiently close to $1$ and the unit normal of $V$ satisfies a $C^{0,α}$ estimate, then $V\cap B_{ρ/2}(x)$ is the graph of a $C^{1,α}$ function with estimates. Furthermore, a similar boundary regularity theorem is true.

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