Locality of the mean curvature of rectifiable varifolds


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Abstract: The aim of this paper is to investigate whether, given two rectifiable k-varifolds in Rn with locally bounded first variations and integer-valued multiplicities, their generalized mean curvatures coincide Hk -almost everywhere on the intersection of the supports of their weight measures. This so-called locality property, which is well-known for classical C2 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.

Mots Clés: Varifolds; Mean curvature; Semicontinuity; Willmore energy; Image reconstruction