Rigidity of Minimal Surfaces with Y-singularities and low Morse Index in $\mathbb{R}^3$

We investigate the geometric constraints imposed by low Morse index on minimal surfaces with Y-singularities, focusing on the classification of those with Morse index one. Our rigidity result establishes a partial uniqueness theorem, highlighting the Y-catenoid as a distinguished example among complete, two-sided minimal surfaces in $\mathbb{R}^3$ with Y-singularities and Morse index one.