Isometric embedding of the 2-sphere with non negative curvature in \(\mathbb{R}^ 3\). (English) Zbl 0833.53049
In 1916 H. Weyl asked whether \(S^2\) equipped with a Riemannian metric \(g\) of positive Gauss curvature \(K(g)\) can be isometrically embedded in \(\mathbb{R}^3\) equipped with the standard metric. Affirmative answers for metrics of various degrees of smoothness were provided by Lewy, Nirenberg, Heinz, Aleksandrov and Pogorelov. Recently J. A. Iaia [Duke Math. J. 67, No. 2, 423-459 (1992; Zbl 0777.53006)]proved the existence of a \(C^{1,1}\) embedding if \(K(g)\) is positive everywhere except at one point \(P\), and \(\Delta_g K\geq 0\) near \(P\). Here the authors extend this result to arbitrary \(C^4\) metrics of nonnegative curvature. The same result was also obtained recently by P. F. Guan and Y. Y. Li [J. Differ. Geom. 39, No. 2, 331-342 (1994; Zbl 0796.53056)]using a somewhat different method.
Reviewer: J.Urbas (Bonn)
MSC:
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
| 35J60 | Nonlinear elliptic equations |
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