# Rectifiability of line defects in liquid crystals with variable degree of orientation

## TL;DR

This paper proves that the defect set of minimizers in liquid crystal models is locally rectifiable, showing each defect curve has finite length, which advances understanding of defect geometry in liquid crystals.

## Contribution

It establishes the rectifiability of defect curves in liquid crystals, building on previous work and simplifying the analysis through blow-up techniques.

## Key findings

- Defect set consists of finite-length curves.
- Each defect curve is of finite length.
- The defect set is locally rectifiable.

## Abstract

In [AHL] Hardt, Lin and the author proved that the defect set of minimizers of the modified Ericksen energy for nematic liquid crystals consists locally of a finite union of isolated points and H\"older continuous curves with finitely many crossings. In this article, we show that each H\"older continuous curve in the defect set is of finite length. Hence, the defect set is locally rectifiable. For the most part, the proof follows the work of De Lellis, Marchese, Spadaro and Valtorta [DLMSV] on harmonic $\mathcal{Q}$-valued maps closely. The blow-up analysis in [AHL] allows us to simplify the covering arguments in [DLMSV] and locally estimate the length of line defects in a geometric fashion.