A Corrected Proof of the Graphical Representation of a Class of Curvature Varifolds by $C^{1,\alpha }$ Multiple Valued Functions

We provide a counter-example to Hutchinson’s original proof of $C^{1,\alpha }$ representation of curvature $m$-varifolds with $L^q$-integrable second fundamental form and $q>m$ in [6]. We also provide an alternative proof of the same result and introduce a method of decomposing varifolds into nested components preserving weakly differentiability of a given function. Furthermore, we prove the structure theorem for curvature varifolds with null second fundamental form which is widely used in the literature.