## 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

### Keywords:

Gauss curvature; isometric embedding
Full Text:

### References:

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