Locality of the mean curvature of rectifiable varifolds
نویسندگان
چکیده
The aim of this paper is to investigate whether, given two rectifiable k-varifolds in R with locally bounded first variations and integer-valued multiplicities, their mean curvatures coincide H-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 multiplicities on their intersection). We also discuss a couple of applications in elasticity and computer vision.
منابع مشابه
Currents and Flat Chains Associated to Varifolds, with an Application to Mean Curvature Flow
We prove under suitable hypotheses that convergence of integral varifolds implies convergence of associated mod 2 flat chains and subsequential convergence of associated integer-multiplicity rectifiable currents. The convergence results imply restrictions on the kinds of singularities that can occur in mean curvature flow.
متن کاملSurface approximation, discrete varifolds, and regularized first variation
Shape visualization and processing are fundamental tasks in many fields from mechanical engineering to physics, biology, chemical engineering, medicine, astronomy, etc., and are the subject of very active research in image processing and computer graphics. For there is a huge variety of applications and a large variety of capture systems, there are many discrete models for representing a shape:...
متن کاملA Generalization of Rellich's Theorem and Regularity of Varifolds Minimizing Curvature a Generalization of Rellich's Theorem and Regularity of Varifolds Minimizing Curvature
We prove a generalization of Rellich's theorem for weakly diieren-tiable functions on curvature varifolds (see deenitions below) and apply it to prove regularity of minimizers of curvature integrals under certain assumptions.
متن کاملThe Nature of Singularities in Mean Curvature Flow of Mean-convex Sets
Let K be a compact subset of R, or, more generally, of an (n+1)-dimensional riemannian manifold. We suppose that K is mean-convex. If the boundary of K is smooth and connected, this means that the mean curvature of ∂K is everywhere nonnegative (with respect to the inward unit normal) and is not identically 0. More generally, it means that Ft(K) is contained in the interior of K for t > 0, where...
متن کاملSingularity Structure in Mean Curvature Flow of Mean Convex Sets
In this note we announce results on the mean curvature flow of mean convex sets in 3-dimensions. Loosely speaking, our results justify the naive picture of mean curvature flow where the only singularities are neck pinches, and components which collapse to asymptotically round spheres. In this note we announce results on the mean curvature flow of mean convex sets; all the statements below have ...
متن کامل