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

Abstract

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 ${\omega }_n^{-1}{\rho }^{-n}\mathrm{area}(V\cap B_{\rho }(x))$ 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.

20 pages