Strong approximation in h-mass of rectifiable currents under homological constraint

TL;DR

This paper proves that rectifiable chains with finite h-mass can be closely approximated by polyhedral chains while preserving boundary constraints, advancing the understanding of approximation in geometric measure theory.

Contribution

It introduces a strong approximation theorem for rectifiable chains under homological constraints using h-mass, with explicit decomposition into polyhedral and rectifiable parts.

Findings

Any rectifiable chain with finite h-mass can be approximated by a polyhedral chain plus a small rectifiable correction.

The approximation preserves the boundary homological constraint exactly.

Provides formulas for the lower semicontinuous envelope of the h-mass functional.

Abstract

Let h : R $\rightarrow$ R+ be a lower semi-continuous subbadditive and even function such that h(0) = 0 and h($\theta$) $\ge$ $\alpha$|$\theta$| for some $\alpha$ > 0. The h-mass of a k-polyhedral chain P =$\sum$j $\theta$j$\sigma$j in R n (0 $\le$ k $\le$ n) is defined as M h (P) := j h($\theta$j) H k ($\sigma$j). If T = $\tau$ (M, $\theta$, $\xi$) is a k-rectifiable chain, the definition extends to M h (T) := M h($\theta$) dH k. Given such a rectifiable flat chain T with M h (T) < $\infty$ and $\partial$T polyhedral, we prove that for every $\eta$ > 0, it decomposes as T = P + $\partial$V with P polyhedral, V rectifiable, M h (V) < $\eta$ and M h (P) < M h (T) + $\eta$. In short, we have a polyhedral chain P which strongly approximates T in h-mass and preserves the homological constraint $\partial$P = $\partial$T. These results are motivated by the study of approximations of M h by smoother functionals but they also provide explicit formulas for the lower semicontinuous envelope of T $\rightarrow$ M h (T) + I $\partial$S ($\partial$T) with respect to the topology of the flat norm.