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Isometric embeddings of surfaces with nonnegative curvature in \(\mathbb{R}^ 3\). (English) Zbl 0777.53006

The main result: Let \(g\in C^ 4\) be a Riemannian metric on \(S^ 2\) with Gauss curvature \(K\) satisfying (i) \(K(P)=0\), \(K(Q)>0\) for \(Q\neq P\); (ii) \(\Delta_ gK(Q)\geq 0\) for all \(Q\in B_ r(P)\) for some \(r>0\). Then there exists a \(C^{1,1}\) isometric embedding \(X:(S^ 2,g)\to (\mathbb{R}^ 3,h)\) where \(h\) is the standard flat metric on \(\mathbb{R}^ 3\). In addition, if \(g\in C^ 5\), \(\limsup_{Q\to P}|\nabla K|^ 2/K=C<\infty\) and \(\liminf_{Q\to P}H(Q)=c>0\), then \(k_ 1\in C^{0,1}\) in a small ball about \(P\) where \(k_ 1\) is the smaller of the two principal curvatures and \(H\) is the mean curvature of the embedding \(X\).

MSC:

53A05 Surfaces in Euclidean and related spaces
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