A curvature estimate for the Weyl problem in the De Sitter space

The problem of isometric embedding of a positively curved 2-sphere in the Euclidean 3-space was considered by Hermann Weyl in 1916 and it's known as the classical Weyl problem.
In this talk we will give an overview of Nirenberg's solution to this problem and where curvature estimates are needed.
Then we will consider the existence of (spacelike) isometric embeddings of a metric on the sphere into de Sitter space, with a suitable curvature restriction.
We show a bound for the mean curvature H of such spacelike hypersurfaces in terms of the scalar curvature, its Laplacian, the dimension and a scaling factor of the ambient space.
The proof uses geometric identities, and the maximum principle for a prescribed symmetric-curvature equation.

This is joint work with Ben Lambert and Wilhelm Klingenberg Jr.