Calculus of Variations and Geometric Measure Theory

G. P. Leonardi - S. Masnou

Locality of the mean curvature of rectifiable varifolds

created by leonardi on 26 Mar 2008
modified on 05 Oct 2010


Published Paper

Inserted: 26 mar 2008
Last Updated: 5 oct 2010

Journal: Adv. Calc. Var.
Volume: 2
Number: 1
Pages: 17-42
Year: 2009


The aim of this paper is to investigate whether, given two rectifiable $k$-varifolds in $\R^n$ with locally bounded first variations and integer-valued multiplicities, their generalized mean curvatures coincide ${\mathcal H}^k$-almost everywhere on the intersection of the supports of their weight measures. This so-called \textit{locality property}, which is well-known for classical $C^2$ surfaces, is far from being obvious in the context of varifolds. We prove that the locality property holds true for integral $1$-varifolds, while for $k$-varifolds, $k>1$, we are able to prove that it is verified under some additional assumptions (local inclusion of the supports and locally constant multiplicity on their intersection). We also discuss a couple of applications in elasticity and computer vision.