A priori bounds for co-dimension one isometric embeddings

Abstract

Let X : ([inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /], g ) → [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] be a C 4 isometric embedding of a C4 metric g of nonnegative sectional curvature on [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] into the Euclidean space [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /]. 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 ([inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="05i" /], 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 [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="06i" /] 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 [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="07i" /] which admits locally convex isometric embedding into [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="08i" /]. Does ([inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="09i" /], g ) then admit a smooth global isometric embedding X : ([inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10i" /], g ) → [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11i" /]?