In 1938, Alexandroff introduced a class of smooth metrics $g$ in the
unit disc $D\subset \mathbb R^2$ such that the Gauss curvature $K$ satisfies
$K>0$ in $D$, $K=0$ and $dK\neq 0$ on $\partial D$ and the total curvature of
$g$ in $D$ is $4\pi$. Alexandroff proved that such metrics are rigid, in the
sense that the isometric embedding of $(D, g)$ in $\mathbb R^3$ is unique up to
rigid body motions if it exists. It is easy to derive necessary conditions for
such metrics to be isometrically embedded in $\mathbb R^3$, among which the
geodesic curvature of the boundary is negative. We will prove that those
necessary conditions are also sufficient.

The proof is based on a discussion of elliptic Monge-Ampere equations which are
degenerate on the boundary. Because of the rigidity, boundary conditions cannot
be described. In fact, there is only one boundary condition which makes this
Monge-Ampere equation solvable.