Quantitative conditions of rectifiability for varifolds

TL;DR

This paper establishes quantitative conditions under which limits of sequences of varifolds are rectifiable, facilitating the approximation of geometric objects like curves and surfaces by discrete models.

Contribution

It introduces a new sequence of functionals that, combined with density estimates, guarantees rectifiability of varifold limits, bridging discrete and continuous geometric analysis.

Findings

Finite energy bounds imply rectifiability of varifold limits.

Uniform density estimates at a scale ensure rectifiability.

Framework supports approximation of geometric sets by discrete objects.

Abstract

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 $\beta_i$, 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, surfaces, or even more general $d$--rectifiable sets minimizing geometrical functionals (like the length for curves or the area for surfaces), can be approximated by "discrete" objects (volumetric approximations, pixelizations, point clouds etc.) minimizing some suitable "discrete" functionals.