Second Order Rectifiability of Integral Varifolds of Locally Bounded First Variation
نویسنده
چکیده
In this work it is shown that for every integral n varifold in an open subset U of Rn+m, n,m ∈ N, of locally bounded first variation there exists a countable collection C of n dimensional submanifolds of U of class C2 such that μ(U ∼ S C) = 0 and for each member M of C ~ Hμ(x) = ~ HM (x) for μ almost all x ∈M.
منابع مشابه
Lower Semicontinuity of the Willmore Functional for Currents
The weak mean curvature is lower semicontinuous under weak convergence of varifolds that is if μk → μ weakly as varifolds then ‖ ~ Hμ ‖Lp(μ)≤ lim infk→∞ ‖ ~ Hμk ‖Lp(μk) . In contrast, if Tk → T weakly as integral currents then μT may not have locally bounded first variation even if ‖ ~ HμTk ‖L∞(μk) is bounded. In 1999, Luigi Ambrosio asked the question whether lower semicontinuity of the weak m...
متن کاملLocality of the mean curvature of rectifiable varifolds
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 o...
متن کاملLorentzian varifolds and applications to closed relativistic strings
We develop a suitable generalization of Almgren’s theory of varifolds in a lorentzian setting, focusing on area, first variation, rectifiability, compactness and closure issues. Motivated by the asymptotic behaviour of the scaled hyperbolic Ginzburg-Landau equations, and by the presence of singularities in lorentzian minimal surfaces, we introduce, within the varifold class, various notions of ...
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 کاملSparse rectifiability and compactness in SBV
We introduce a notion of sparse rectifiability, stronger than that of uniform rectifiability. As applications we derive, firstly, results ensuring the convergence of the total variation measures |μ| subject to the weak* convergence of the sparsely rectifiable Radon measures μ. Secondly, we apply sparse rectifiability to derive compactness results for special functions of bounded variation (SBV)...
متن کامل