| A Priori Bounds For Co-Dimension One Isometric Embeddings (1999) | |||||||||||||
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| . Let X : (S n ; g) ! R n+1 be a C 4 isometric embedding of a C 4 metric g of non-negative sectional curvature on S n into the Euclidean space R 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 (S 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 C 2;ff 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 ffl ? 0, the set of metrics of non-negative sectional curvature and scalar curvature bounded below by ffl which are isometrically embedable in Euclidean space R n+1 is closed in the Holder space C 4;ff , 0 ! f... | |||||||||||||
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