


The
Christoffel problem and two analogs of the Minkowski problem in Riemannian
space
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 Gdeformations preserving the sum of
principal radii of curvature for every point of surface in Riemannian space.
Gdeformation 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 Gdeformation; 2) the problem of
construction the deformations preserving the area of each arbitrary region of
surface and condition of Gdeformation.
Keywords: Mathematics  Differential Geometry, Mathematics  Analysis of
PDEs, 53C20, 53C21
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