# Uniqueness of planar tangent maps in the modified Ericksen model

## TL;DR

This paper proves the uniqueness of tangent maps at singularities for energy-minimizing nematic liquid crystal models in planar domains, using advanced monotonicity and blow-up techniques.

## Contribution

It establishes the uniqueness of homogeneous blow-up limits for modified Ericksen energy minimizers in 2D, extending techniques from minimal surface and harmonic map theory.

## Key findings

- Uniqueness of tangent maps at singularities in the modified Ericksen model.
- Application of Weiss monotonicity formula to liquid crystal energy minimizers.
- Extension of classical blow-up methods to nematic liquid crystal models.

## Abstract

We prove the uniqueness of homogeneous blow-up limits of maps minimizing the modified Ericksen energy for nematic liquid crystals in a planar domain. The proof is based on the Weiss monotonicity formula, and a blow-up argument, originally due to Allard and Almgren \cite{AA} for minimal surfaces, and L. Simon \cite{SL} for energy-minimizing maps into analytic targets, which exploits the integrability of certain Jacobi fields.