@article{Lin1985,
  author = {Lin, Chang-Shou},
  title = {The local isometric embedding in {$\mathbb{R}^3$} of 2-dimensional {Riemannian} manifolds with nonnegative curvature},
  journal = {Journal of Differential Geometry},
  volume = {21},
  number = {2},
  year = {1985},
  pages = {213--230},
  doi = {10.4310/jdg/1214439563}
}

@article{Lin1986,
  author = {Lin, Chang-Shou},
  title = {The local isometric embedding in {$\mathbb{R}^3$} of two-dimensional {Riemannian} manifolds with {Gaussian} curvature changing sign cleanly},
  journal = {Communications on Pure and Applied Mathematics},
  volume = {39},
  number = {6},
  year = {1986},
  pages = {867--887},
  doi = {10.1002/cpa.3160390607}
}

@article{NakamuraMaeda1989,
  author = {Nakamura, Gen and Maeda, Yoshiaki},
  title = {Local smooth isometric embeddings of low-dimensional {Riemannian} manifolds into {Euclidean} spaces},
  journal = {Transactions of the American Mathematical Society},
  volume = {313},
  number = {1},
  year = {1989},
  pages = {1--51},
  doi = {10.1090/S0002-9947-1989-0992597-8}
}

@article{Nash1954,
  author = {Nash, John},
  title = {{$C^1$} isometric imbeddings},
  journal = {Annals of Mathematics},
  series = {2},
  volume = {60},
  number = {3},
  year = {1954},
  pages = {383--396},
  doi = {10.2307/1969840}
}

@article{Kuiper1955,
  author = {Kuiper, Nicolaas H.},
  title = {On {$C^1$}-isometric imbeddings. {I}},
  journal = {Nederl. Akad. Wetensch. Proc. Ser. A},
  volume = {58},
  year = {1955},
  pages = {545--556}
}

@article{Khuri2007Counterexamples,
  author = {Khuri, Marcus A.},
  title = {Counterexamples to the Local Solvability of {Monge--Amp\`ere} Equations in the Plane},
  journal = {Communications in Partial Differential Equations},
  volume = {32},
  number = {4},
  year = {2007},
  pages = {665--674},
  doi = {10.1080/03605300600635061}
}

@book{Yau,
  author    = {Schoen, Richard and Yau, Shing-Tung},
  title     = {Lectures on Differential Geometry},
  series    = {Conference Proceedings and Lecture Notes in Geometry and Topology},
  volume    = {1},
  publisher = {International Press},
  address   = {Cambridge, MA},
  year      = {1994},
  isbn      = {1-57146-012-8},
  note      = {Chapter IX, Problem 22, p.~375},
  url       = {https://www.intlpress.com/site/pub/files/preview/bookpubs/00000308.pdf}
}

@article{Pogorelov1971,
  author  = {Pogorelov, A. V.},
  title   = {An example of a two-dimensional {Riemannian} metric admitting no local realization in {$E^3$}},
  journal = {Doklady Akademii Nauk SSSR},
  volume  = {198},
  number  = {1},
  year    = {1971},
  pages   = {42--43},
  note    = {In Russian; English translation in Soviet Mathematics Doklady 12 (1971), 729--730},
  url     = {https://www.mathnet.ru/eng/dan36135}
}

@article{NY2008,
  author  = {Nadirashvili, Nikolai and Yuan, Yu},
  title   = {Improving {Pogorelov}'s isometric embedding counterexample},
  journal = {Calculus of Variations and Partial Differential Equations},
  volume  = {32},
  year    = {2008},
  pages   = {319--323},
  doi     = {10.1007/s00526-007-0140-7},
  url     = {https://sites.math.washington.edu/~yuan/papers/2008CVPDE.pdf}
}

@misc{NY2002,
  author        = {Nadirashvili, Nikolai and Yuan, Yu},
  title         = {Counterexamples for Local Isometric Embedding},
  year          = {2002},
  eprint        = {math/0208127},
  archivePrefix = {arXiv},
  primaryClass  = {math.DG},
  note          = {Preprint, arXiv:math/0208127v1, 16 August 2002},
  url           = {https://arxiv.org/abs/math/0208127v1}
}

@article{Khuri2009,
  author  = {Khuri, Marcus A.},
  title   = {On the local solvability of {Darboux}'s equation},
  journal = {Discrete and Continuous Dynamical Systems},
  year    = {2009},
  note    = {Supplement},
  pages   = {451--456},
  doi     = {10.3934/proc.2009.2009.451},
  url     = {https://www.math.stonybrook.edu/~khuri/Khuri_AIMS_Darboux.pdf}
}

@article{HanKhuri2010,
  author        = {Han, Qing and Khuri, Marcus},
  title         = {On the local isometric embedding in {$\mathbb{R}^3$}
                   of surfaces with {Gaussian} curvature of mixed sign},
  journal       = {Communications in Analysis and Geometry},
  volume        = {18},
  number        = {4},
  year          = {2010},
  pages         = {649--704},
  doi           = {10.4310/CAG.2010.v18.n4.a2},
  eprint        = {1009.6214},
  archivePrefix = {arXiv},
  url           = {https://arxiv.org/abs/1009.6214}
}

@article{Lin2015,
  author        = {Lin, Tsung-Yin},
  title         = {On the local isometric embedding in {$\mathbb{R}^3$}
                   of surfaces with zero sets of {Gaussian} curvature
                   forming cusp domains},
  journal       = {Communications in Partial Differential Equations},
  volume        = {41},
  number        = {10},
  year          = {2016},
  pages         = {1553--1592},
  doi           = {10.1080/03605302.2016.1227336},
  eprint        = {1510.07505},
  archivePrefix = {arXiv},
  url           = {https://arxiv.org/abs/1510.07505}
}

@misc{Ghomi2019,
  author = {Ghomi, Mohammad},
  title  = {Open problems in geometry of curves and surfaces},
  year   = {2019},
  note   = {Author-hosted survey, revised 2 September 2019; Section 1.5},
  url    = {https://ghomi.math.gatech.edu/Papers/op.pdf}
}

@article{Janet1926,
  author  = {Janet, Maurice},
  title   = {Sur la possibilit{\'e} de plonger un espace riemannien donn{\'e}
             dans un espace euclidien},
  journal = {Annales de la Soci{\'e}t{\'e} Polonaise de Math{\'e}matique},
  volume  = {5},
  year    = {1926},
  pages   = {38--43},
  url     = {https://rcin.org.pl/dlibra/publication/31746/edition/15034}
}

@article{Cartan1927,
  author  = {Cartan, {\'E}lie},
  title   = {Sur la possibilit{\'e} de plonger un espace riemannien donn{\'e}
             dans un espace euclidien},
  journal = {Annales de la Soci{\'e}t{\'e} Polonaise de Math{\'e}matique},
  volume  = {6},
  year    = {1927},
  pages   = {1--7},
  url     = {https://rcin.org.pl/dlibra/publication/32955/edition/18106}
}

@article{Yau2000,
  author  = {Yau, Shing-Tung},
  title   = {Review of geometry and analysis},
  journal = {Asian Journal of Mathematics},
  volume  = {4},
  number  = {1},
  year    = {2000},
  pages   = {235--278},
  doi     = {10.4310/AJM.2000.v4.n1.a16},
  url     = {https://archive.intlpress.com/site/pub/files/_fulltext/journals/ajm/2000/0004/0001/AJM-2000-0004-0001-a016.pdf}
}