Immersions isométriques elliptiques et courbes pseudo-holomorphes. (Elliptic isometric immersions and pseudo-holomorphic curves).

*(French)*Zbl 0682.53063In this article result of A. V. Pogorelov [from “Extrinsic geometry of convex surfaces”, translated from the Russian by Israel Program for Scientific Translations, translations of Mathematical Monographs, Am. Math. Soc. VI (1973; Zbl 0311.53067)] are generalized and some of the proofs are shortened.

Let S and M be oriented Riemannian manifolds of dimension 2 and 3, respectively. Then a suitable open part O of the 1-jet-bundle of isometric immersions f: \(S\to M\) can be equipped with a canonical almost complex structure J. If f is elliptic \((=locally\) convex), i.e. the determinant of its shape operator is strictly positive, then its 1-jet is a pseudo-holomorphic curve in (O,J). Using Gromov’s generalization of Schwarz’s Lemma the author derives theorems on the convergence of sequences of pseudo-holomorphic maps which, applied to elliptic isometric immersions, give the basic tool of the article. First he can simplify the proof of Pogorelov’s theorem which states: If the sectional curvature \(K_ S\) and \(K_ M\) of S and M satisfy \(K_ M\leq K_ 0<K_ S\) with some \(K_ 0\in {\mathbb{R}}\) and S is homeomorphic to a sphere, then S can isometrically be immersed into M. Next he generalizes this result in the case that M is simply connected and S is metrically complete and homeomorphic to an open part of a sphere; under the same curvature condition then even S can isometrically be embedded into M.

Using pseudo-holomorphic curves the author can also shorten the proof of Pogorelov’s theorem on the infinitesimal rigidity of elliptic isometric immersions f: \(S\to M\) with compact S.

Let S and M be oriented Riemannian manifolds of dimension 2 and 3, respectively. Then a suitable open part O of the 1-jet-bundle of isometric immersions f: \(S\to M\) can be equipped with a canonical almost complex structure J. If f is elliptic \((=locally\) convex), i.e. the determinant of its shape operator is strictly positive, then its 1-jet is a pseudo-holomorphic curve in (O,J). Using Gromov’s generalization of Schwarz’s Lemma the author derives theorems on the convergence of sequences of pseudo-holomorphic maps which, applied to elliptic isometric immersions, give the basic tool of the article. First he can simplify the proof of Pogorelov’s theorem which states: If the sectional curvature \(K_ S\) and \(K_ M\) of S and M satisfy \(K_ M\leq K_ 0<K_ S\) with some \(K_ 0\in {\mathbb{R}}\) and S is homeomorphic to a sphere, then S can isometrically be immersed into M. Next he generalizes this result in the case that M is simply connected and S is metrically complete and homeomorphic to an open part of a sphere; under the same curvature condition then even S can isometrically be embedded into M.

Using pseudo-holomorphic curves the author can also shorten the proof of Pogorelov’s theorem on the infinitesimal rigidity of elliptic isometric immersions f: \(S\to M\) with compact S.

Reviewer: H.Reckziegel

##### MSC:

53C40 | Global submanifolds |

53C45 | Global surface theory (convex surfaces à la A. D. Aleksandrov) |

53C55 | Global differential geometry of Hermitian and Kählerian manifolds |