On Weyl's embedding problem in Riemannian manifolds

Understand

We consider a priori estimates of Weyl's embedding problem of $(\mathbb{S}^2, g)$ in general $3$-dimensional Riemannian manifold $(N^3, \bar g)$.

• 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 $D^2g$ Dini continuous.