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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

Marcus A. Khuri Stanford University

Differential Geometry mathscidoc:1609.10022

Journal of Differential Geometry, 76, (2), 249-291, 2007
We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge- Amp`ere type. These two problems are: the local isometric embedding problem for two-dimensional Riemannian manifolds, and the problem of locally prescribed Gaussian curvature for surfaces in R3. We prove a general local existence result for a large class of Monge-Amp`ere equations in the plane, and obtain as corollaries the existence of regular solutions to both problems, in the case that the Gaussian curvature vanishes to arbitrary finite order on a single smooth curve.
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@inproceedings{marcus2007the,
  title={The local isometric embedding in $\ mathbb {R}^ 3$ of two-dimensional Riemannian manifolds with Gaussian curvature changing sign to finite order on a curve},
  author={Marcus A. Khuri},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160908201117993448679},
  booktitle={Journal of Differential Geometry},
  volume={76},
  number={2},
  pages={249-291},
  year={2007},
}
Marcus A. Khuri. The local isometric embedding in $\ mathbb {R}^ 3$ of two-dimensional Riemannian manifolds with Gaussian curvature changing sign to finite order on a curve. 2007. Vol. 76. In Journal of Differential Geometry. pp.249-291. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160908201117993448679.
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