A corrected proof of the graphical representation of a class of curvature varifolds by C 1,α multiple valued functions

Abstract We provide a counter-example to Hutchinson’s original proof of C 1, α C^{1, } representation of curvature m -varifolds with L q L^{q} -integrable second fundamental form and q > m q>m in J. E. Hutchinson, C 1, α C^{1, } multiple function regularity and tangent cone behaviour for varifolds with second fundamental form in L p L^{p}, Geometric Measure Theory and the Calculus of Variations (Arcata 1984), Proc. Sympos. Pure Math. 44, American Mathematical Society, Providence 1986, 281–306. 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.