Isometric embedding of the 2-sphere with non negative curvature in \(\mathbb{R}^ 3\)
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 \textit{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 \textit{P. F. Guan} and \textit{Y. Y. Li} [J. Differ. Geom. 39, No. 2, 331-342 (1994; Zbl 0796.53056)]\ using a somewhat different method.
scientific article; zbMATH DE number 795024