I am curious about the following existence result.
A compact connected set $\omega$ in $\mathbb{R}^2$ is given, along with a symmetric positive definite matrix field $[a_{\alpha\beta}]$ and a symmetric matrix field $[b_{\alpha\beta}]$ over $\omega$, all sufficiently smooth, where the lowercase Greek indices $\alpha, \beta$ etc. assume value from the set $\{1,2\}$. Then what are the conditions on $[a_{\alpha\beta}]$ and $[b_{\alpha\beta}]$ such that there exists a sufficiently smooth isometric embedding $\mathbf{r}:\omega\to\mathbb{R}^3$ which satisfies $\mathbf{r}_{,\alpha}\boldsymbol{\cdot}\mathbf{r}_{,\beta}=a_{\alpha\beta}$ and $-\mathbf{n}_{,\alpha}\boldsymbol{\cdot}\mathbf{r}_{,\alpha}=b_{\alpha\beta}$ everywhere on $\omega$, where $\mathbf{n}:=\frac{\mathbf{r}_{,1}\times\mathbf{r}_{,2}}{|\mathbf{r}_{,1}\times\mathbf{r}_{,2}|}$? Here, $\boldsymbol{\cdot}$, $|\cdot|$ and $\times$ denote the Euclidean inner product, the Euclidean norm and the vector cross product in $\mathbb{R}^3$, respectively.
A local necessary and sufficient condition is known. If $\omega$ is simply-connected, then the conditions on $a_{\alpha\beta}$ and $b_{\alpha\beta}$ are the following Gauss and Codazzi-Mainardi euqations:
$$ K_{1212}=b^2_{12}-b_{11}b_{12}$$ $$\partial_1 b_{\alpha 2}- \partial_2 b_{\alpha 1}=0$$. Here, $K_{\alpha\beta\mu\nu}$ denotes the covariant components of the Riemannian curvature of the metric $a_{\alpha\beta}$ and $\partial$ denotes the covariant derivative with respect to the Levi-Civita connection of $a_{\alpha\beta}$.
Is there anything known about the global existence result? What if $\omega$ is multiply-connected and/or non-orientable?
References would be greatly appreciated!