Communications in Analysis and Geometry

Volume 18 (2010)

Number 4

On the local isometric embedding in $\mathbb{R}^{3}$ of surfaces with Gaussian curvature of mixed sign

Pages: 649 – 704



Qing Han (Department of Mathematics, University of Notre Dame, Notre Dame, Indiana)

Marcus Khuri (Department of Mathematics, Stony Brook University, Stony Brook, New York)


We study the old problem of isometrically embedding a two-dimensional Riemannian manifold into Euclidean three-space. It is shown that if the Gaussian curvature vanishes to finite order and its zero set consists of two Lipschitz curves intersecting transversely at a point, then local sufficiently smooth isometric embeddings exist.

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