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The Christoffel problem and two analogs of the Minkowski problem in Riemannian space

Bodrenko, Andrei I.

eprint arXiv:0709.0158

Author finds the solutions of the Christoffel problem for open and closed surfaces in Riemannian space. The Christoffel problem is reduced to the problem of construction the continuous G-deformations preserving the sum of principal radii of curvature for every point of surface in Riemannian space. G-deformation transfers every normal vector of surface in parallel along the path of the translation for each point of surface. The following analogs of the Minkowski problem for open and closed surfaces in Riemannian space are being considered in this article: 1) the problem of construction the surface with prescribed mean curvature and condition of G-deformation; 2) the problem of construction the deformations preserving the area of each arbitrary region of surface and condition of G-deformation.

Keywords: Mathematics - Differential Geometry, Mathematics - Analysis of PDEs, 53C20, 53C21

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