A priori bounds for co-dimension one isometric embeddings

Let X: (double-struck S signⁿ, g) → ℝⁿ⁺¹ be a C⁴ isometric embedding of a C⁴ metric g of nonnegative sectional curvature on double-struck S signⁿ into the Euclidean space ℝⁿ⁺¹. 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 (double-struck S signⁿ, 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 nonnegative curvature and n = 2. Using C^2,α 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 ∈ > 0, the set of metrics of nonnegative sectional curvature and scalar curvature bounded below by ∈ which are isometrically embedable in Euclidean space ℝⁿ⁺¹ is closed in the Hölder space C^4,α, 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 nonnegative sectional curvature and positive scalar curvature on double-struck S signⁿ which admits locally convex isometric embedding into ℝⁿ⁺¹. Does (double-struck S signⁿ, g) then admit a smooth global isometric embedding X: (double-struck S signⁿ, g) → ℝⁿ⁺¹ ?