This question has [also been posted on MSE][1], but maybe here is the right place to post it.

Is it true that if $D$ is a Riemannian $2$-disk having constant Gaussian curvature equal to $1$ and whose boundary has constant geodesic curvature, then $D$ is isometric to some geodesic ball of the unit sphere $\mathbb{S}^2 \subset \mathbb{R}^3$? I strongly suspect so, but I couldn't find a reasonable argument. 


  [1]: https://math.stackexchange.com/questions/3986936/characterization-of-disks-of-constant-curvature-and-whose-boundaries-have-consta