A free boundary isometric embedding problem in the unit ball

Abstract

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 R 3 such that the image of the boundary meets the unit sphere 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. * is non-empty but we would of course like to show that G k,α * = G k,α . In order to do so, we equip G k,α with the C k,α (D, Sym(R 2 ))−topology and show that G k,α is path-connected, open and closed. It actually turns out that G k,α is not only path-connected but that the paths can be chosen to be analytic maps from the unit interval to G k,α .