Geodesic
On~$\Sigma$~-- realizations of metrics of positive curvature
Sbornik. Mathematics, Tome 45 (1983) no. 4, pp. 515-525

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A metric $ds^2$ admits a $\Sigma$-realization if there is a realization of it in $E^3$ in the form of a surface whose boundary lies on a given surface $\Sigma$. This paper proves the existence of $\Sigma$-realizations of a certain class of metrics of positive curvature for surfaces of quite general form, and describes a number of possible $\Sigma$-realizations of the given metric. The proof is based on a consideration of a nonlinear boundary-value problem for immersion equations. Bibliography: 3 titles. 
@article{SM_1983_45_4_a7,
     author = {V. T. Fomenko},
     title = {On~$\Sigma$~-- realizations of metrics of positive curvature},
     journal = {Sbornik. Mathematics},
     pages = {515--525},
     publisher = {mathdoc},
     volume = {45},
     number = {4},
     year = {1983},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1983_45_4_a7/}
}
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V. T. Fomenko. On~$\Sigma$~-- realizations of metrics of positive curvature. Sbornik. Mathematics, Tome 45 (1983) no. 4, pp. 515-525. http://geodesic.mathdoc.fr/item/SM_1983_45_4_a7/