Geodesic
Locally Euclidean Metrics with a~Given Geodesic Curvature of the Boundary
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometry, topology, and mathematical physics. II, Tome 266 (2009), pp. 218-226.

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The problem of reconstructing a locally Euclidean metric on a disk from the geodesic curvature of the boundary given in the sought metric is considered. This problem is an analog and a generalization of the classical problem of finding a closed plane curve from its curvature given as a function of the arc length. The solution of this problem in our approach can be interpreted as finding a plane domain with the standard Euclidean metric whose boundary has a given geodesic curvature. 
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     author = {I. Kh. Sabitov},
     title = {Locally {Euclidean} {Metrics} with {a~Given} {Geodesic} {Curvature} of the {Boundary}},
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I. Kh. Sabitov. Locally Euclidean Metrics with a~Given Geodesic Curvature of the Boundary. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometry, topology, and mathematical physics. II, Tome 266 (2009), pp. 218-226. http://geodesic.mathdoc.fr/item/TM_2009_266_a11/

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