Geodesic
The problem of recovering a surface by the given external curvature and solutions of the Monge--Amp\`ere equation
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential equations, geometry, and topology, Tome 201 (2021), pp. 123-131

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In this paper, we generalize the concept of the spherical mapping of a surface in Euclidean space. The normal mapping of a surface introduced by I. Ya. Bakelman is a special case of the generalized curvature. We prove general properties of the generalized curvature and special properties of the generalized curvature extended to a hyperbolic cylinder. Using these properties, we prove the existence and uniqueness of a solution of the Monge–Ampère equation in a multiply connected domain.
Keywords: spherical mapping, external curvature, normal mapping, generalized conditional curvature, hyperbolic cylinder, multiply connected domain. 
@article{INTO_2021_201_a10,
     author = {A. Artikbaev and N. M. Ibodullaeva},
     title = {The problem of recovering a surface by the given external curvature and solutions of the {Monge--Amp\`ere} equation},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {123--131},
     publisher = {mathdoc},
     volume = {201},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2021_201_a10/}
}
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A. Artikbaev; N. M. Ibodullaeva. The problem of recovering a surface by the given external curvature and solutions of the Monge--Amp\`ere equation. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential equations, geometry, and topology, Tome 201 (2021), pp. 123-131. http://geodesic.mathdoc.fr/item/INTO_2021_201_a10/