A Priori Bounds for Co-dimension One Isometric Embeddings

Let X : (S, g) → R be a C isometric embedding of a C 4 metric g of non-negative sectional curvature on S into the Euclidean space R. We prove a priori bounds for the trace of the second fundamental form H , in terms of the scalar curvature R of g, and the diameter d of the space (S, g). These estimates give a bound on the extrinsic geometry in terms of intrinsic quantities. They generalize estimates originally obtained by Weyl for the case n = 2 and positive curvature, and then by P. Guan and the first author for non-negative curvature and n = 2. Using C interior estimates of Evans and Krylov for concave fully nonlinear elliptic partial differential equations, these bounds allow us to obtain the following convergence theorem: For any ǫ > 0, the set of metrics of non-negative sectional curvature and scalar curvature bounded below by ǫ which are isometrically embedable in Euclidean space R is closed in the Hölder space C, 0 < α < 1. These results are obtained in an effort to understand the following higher dimensional version of the Weyl embedding problem which we propose: Suppose that g is a smooth metric of non-negative sectional curvature and positive scalar curvature on S which is locally isometrically embeddable in R. Does (S, g) then admit a smooth global isometric embedding X : (S, g)→ R?