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

A free boundary isometric embedding problem in the unit ball