Eigenvalue problems have played a pivotal role in the study of minimal surface theory. In this talk, I will provide an overview, starting with basic concepts and progressing to recent development. Specifically, I will explain free boundary minimal surfaces in the unit ball and their connection to eigenvalue problems. Key conjectures in this area will be highlighted, along with a discussion of my recent contribution.

We first prove a Boundary Schwarz lemma for holomorphic disks on the unit ball in $\mathbb{C}^n$. Further by using a Schwarz lemma for minimal conformal disks of Forstnerič and Kalaj, we prove the boundary Schwarz lemma for such minimal disks.