On Weyl's embedding problem in Riemannian manifolds

We consider a priori estimates of Weyl’s embedding problem of $($\mathbb{S}$^2, g)$ingeneral$3$-dimensionalRiemannianmanifold$(N^3, \bar g)$. We establish interior $C^2$ estimate under natural geometric assumption. Together with a recent work by Li and Wang, we obtain an isometric embedding of $({\mathbb{S}}^2,g)$ in Riemannian manifold. In addition, we reprove Weyl’s embedding theorem in space form under the condition that $g\in C^2$ with $D^{2}g$ Dini continuous.