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**The local isometric embedding in \(\mathbb R^3\) of two-dimensional Riemannian manifolds with Gaussian curvature changing sign to finite order on a curve.**
*(English)*
Zbl 1130.53017

It is known since a long time that a two-dimensional Riemannian manifold \((M^2, ds^2)\) can be isometrically embedded in \({\mathbb R}^3\) provided that \(ds^2\) is analytic or of Gaussian curvature \(K \neq 0\). For metrics with low regularity the embedding is not always possible, some counterexamples are known. In the paper, which is a revised part of the dissertation supervised by J. Kazdan, the author considers Riemannian manifolds in a neighbourhood of a point at which \(K = 0\). He proves theorems showing the existence of the embedding under appropriate assumptions on the smoothness of \(ds^2\) (\(ds^2 \in C^r\) where \(r \geq 60\) and \(K\) vanishes to a finite order along a single curve passing through the origin). Another (closely related) problem treated in this paper is the existence of a piece of a surface in \({\mathbb R}^3\) under the assumption that its Gaussian curvature is prescribed (\(K \in C^r\), \(r \geq 58\), and \(K\) vanishes to a finite order along a curve). In order to solve the problems the author reduce them to specific equations of Monge-Amperé type. The paper is concerned with a detailed analysis of the local solvability of such equations. To construct local solutions the author applies the Nash-Moser iteration procedure.

Reviewer: Jan L. Cieśliński (Białystok)

### MSC:

53B25 | Local submanifolds |

35A07 | Local existence and uniqueness theorems (PDE) (MSC2000) |

35M10 | PDEs of mixed type |

53A05 | Surfaces in Euclidean and related spaces |

53B20 | Local Riemannian geometry |