The weak convergence of varifolds generated by rectifiable flat G-chains

Non-unique conical and non-conical tangents to rectifiable stationary varifolds in R^4

We construct a rectifiable stationary 2-varifold in R^4 with non-conical, and hence non-unique, tangent varifold at a point. This answers a question of L. Simon (Lectures on geometric measure theory, 1983, p. 243) and provides a new example for a related question of W.K. Allard (On the first variation of a varifold, Ann. of Math., 1972, p. 460). There is also a (rectifiable) stationary 2-varifold in R^4 that has more than one conical tangent varifold at a point. keywords: stationary varifold, varifold tangent, tangent cone, non-unique, non-conical, minimal surface, regularity

Quantitative conditions of rectifiability for varifolds

Our purpose is to state quantitative conditions ensuring the rectifiability of a $d$--varifold $V$ obtained as the limit of a sequence of $d$--varifolds $(V_i)_i$ which need not to be rectifiable. More specifically, we introduce a sequence $\left\lbrace \mathcal{E}_i \right\rbrace_i$ of functionals defined on $d$--varifolds, such that if $\displaystyle \sup_i \mathcal{E}_i (V_i) < +\infty$ and $V_i$ satisfies a uniform density estimate at some scale $β_i$, then $V = \lim_i V_i$ is $d$--rectifiable. \noindent The main motivation of this work is to set up a theoretical framework where curves, su…

数据采样于可测集上的变分量估计

A varifold-type estimation for data sampled on a rectifiable set

本文提出在统计框架下，基于核方法估计变分结构，研究了在高维空间中从底层形状S获得的i.i.d样本的变分结构推断，考虑了密度θ的非均匀性及S的分段C^{1,a}正则性。

Rectifiability of flat chains

We prove (without using Federer's structure theorem) that a finite-mass flat chain over any coefficient group is rectifiable if and only if almost all of its 0-dimensional slices are rectifiable. This implies that every flat chain of finite mass and finite size is rectifiable. It also leads to a simple necessary and sufficient condition on the coefficient group in order for every finite-mass flat chain to be rectifiable.

Rectifiability of flat chains in Banach spaces with coefficients in $Z_p$

Aim of this paper is a finer analysis of the group of flat chains with coefficients in $Z_p$ introduced in a recent paper by Ambrosio-Katz, by taking quotients of the group of integer rectifiable currents, along the lines of the the Ziemer and Federer approach. We investigate the typical questions of the theory of currents, namely rectifiability of the measure-theoretic support and boundary rectifiability. Our main result can also be interpreted as a closure theorem for the class of integer rectifiable currents with respect to a (much) weaker convergence, induced by flat distance mod $p$, and …

Subsets of rectifiable curves in Banach spaces II: universal estimates for almost flat arcs

We prove that in any Banach space the set of windows in which a rectifiable curve resembles two or more straight line segments is quantitatively small with constants that are independent of the curve, the dimension of the space, and the choice of norm. Together with Part I, this completes the proof of the necessary half of the Analyst's Traveling Salesman theorem with sharp exponent in uniformly convex spaces.