## Abstract

Let X: (double-struck S sign^{n}, g) → ℝ^{n+1} be a C^{4} isometric embedding of a C^{4} metric g of nonnegative sectional curvature on double-struck S sign^{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 (double-struck S sign^{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 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 ℝ^{n+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 sign^{n} which admits locally convex isometric embedding into ℝ^{n+1}. Does (double-struck S sign^{n}, g) then admit a smooth global isometric embedding X: (double-struck S sign^{n}, g) → ℝ^{n+1} ?

Original language | English |
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Pages (from-to) | 945-965 |

Number of pages | 21 |

Journal | American Journal of Mathematics |

Volume | 121 |

Issue number | 5 |

DOIs | |

State | Published - Oct 1999 |

Externally published | Yes |