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

TL;DR

This paper corrects and improves upon Hutchinson's proof regarding the graphical representation of curvature varifolds, introduces a new decomposition method, and proves a key structure theorem for varifolds with null second fundamental form.

Contribution

It provides a corrected proof, an innovative decomposition technique, and a structure theorem for specific curvature varifolds, advancing the mathematical understanding of varifold regularity.

Findings

Counter-example to Hutchinson's original proof

Alternative proof of $C^{1,eta}$ representation

Structure theorem for curvature varifolds with null second fundamental form

Abstract

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.