A free boundary isometric embedding problem in the unit ball

In this article, we study a free boundary isometric embedding problem for abstract Riemannian two-manifolds with the topology of the disc. Under the assumption of positive Gauss curvature and geodesic curvature of the boundary being equal to one, we show that any such disc may be isometrically embedded into the Euclidean three space $\mathbb{R}^3$ such that the image of the boundary meets the unit sphere $\mathbb{S}^2$ orthogonally. Moreover, we also show that the embedding is unique up to rotations and reflections through planes containing the origin. Finally, we define a new Brown-York type quasi-local mass for certain free boundary surfaces and discuss its positivity.