# Properties of soliton surfaces associated with integrable $\mathbb{C}P^{N-1}$ sigma models

## TL;DR

This paper explores the geometric properties of soliton surfaces linked to integrable $ ext{CP}^{N-1}$ sigma models, revealing new relations among different immersion formulas and their unified surface representation.

## Contribution

It introduces new properties of projectors and establishes explicit analytical connections among three immersion formulas for soliton surfaces.

## Key findings

- New properties of projectors onto one-dimensional subspaces
- Relations among generalized Weierstrass, Sym-Tafel, and Fokas-Gel'fand formulas
- Unified surface parametrization via specific conditions

## Abstract

We investigate certain properties of $\mathfrak{su}(N)$-valued two-dimensional soliton surfaces associated with the integrable $\mathbb{C}P^{N-1}$ sigma models constructed by the orthogonal rank-one Hermitian projectors, which are defined on the two-dimensional Riemann sphere with finite action functional. Several new properties of the projectors mapping onto one-dimensional subspaces as well as their relations with three mutually different immersion formulas, namely, the generalized Weierstrass, Sym-Tafel and Fokas-Gel'fand have been discussed in detail. Explicit connections among these three surfaces are also established by purely analytical descriptions and, it is demonstrated that the three immersion formulas actually correspond to the single surface parametrized by some specific conditions.