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 consider
(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.