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

We provide a counter-example to Hutchinson’s original proof of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>C</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>α</m:mi> </m:mrow> </m:msup> </m:math> {C1,α} representation of curvature m -varifolds with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mi>q</m:mi> </m:msup> </m:math> {Lq} -integrable second fundamental form and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>q</m:mi> <m:mo>></m:mo> <m:mi>m</m:mi> </m:mrow> </m:math> {q>m} in [J. E. Hutchinson, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>C</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>α</m:mi> </m:mrow> </m:msup> </m:math> {C1,α} multiple function regularity and tangent cone behaviour for varifolds with second fundamental form in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msup> </m:math> {Lᵖ} , 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.