A priori bounds for co-dimension one isometric embeddings

Abstract

We prove a priori bounds for the trace of the second fundamental form of a $C^4$ isometric embedding into $R^{n+1}$ of a metric $g$ of non-negative sectional curvature on $S^n$, in terms of the scalar curvature, and the diameter of $g$. 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 $C^{2,α}$ 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 $ε>0$, the set of metrics of non-negative sectional curvature and scalar curvature bounded below by $ε$ 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: \emph{Suppose that $g$ is a smooth metric of non-negative sectional curvature and positive scalar curvature on §^n$ which is locally isometrically embeddable in $R^{n+1}$. Does $(S^n,g)$ then admit a smooth global isometric embedding into $R^{n+1}$?}