A free boundary isometric embedding problem in the unit ball

TL;DR

This paper proves that certain positively curved Riemannian discs with boundary conditions can be uniquely embedded into three-dimensional space with a free boundary on the unit sphere, and introduces a new quasi-local mass concept.

Contribution

It establishes existence and uniqueness of free boundary isometric embeddings for specific Riemannian discs and proposes a novel Brown-York type quasi-local mass.

Findings

Existence of isometric embeddings into D space with free boundary conditions.

Uniqueness of embeddings up to orthogonal transformations.

Introduction of a new quasi-local mass with positivity properties.

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 $\mathbb{R}^3$ 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.