| A PRIORI BOUNDS FOR CO-DIMENSION ONE ISOMETRIC EMBEDDINGS (2008) | |||||||||||||
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| Abstract. Let X: ( � n, g) → � n+1 be a � 4 isometric embedding of a � 4 metric g of non-negative sectional curvature on � n into the Euclidean space � n+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 ( � n, 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 � 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 ɛ> 0, the set of metrics of non-negative sectional curvature and scalar curvature bounded below by ɛ which are isometrically embedable in Euclidean space � n+1 is closed in the Hölder space � 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 non-negative sectional curvature and positive scalar curvature on � n which is locally isometrically embeddable in � n+1. Does ( � n, g) then admit a smooth global isometric embedding X: ( � n, g) → � n+1? 1. | |||||||||||||
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