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

TL;DR

This paper addresses the Christoffel problem and introduces two Minkowski problem analogs in Riemannian space, focusing on constructing surfaces with specific curvature and deformation-preserving properties.

Contribution

It provides solutions to the Christoffel problem in Riemannian space and formulates two new Minkowski problem analogs involving G-deformations.

Findings

Solutions to the Christoffel problem for surfaces in Riemannian space.

Formulation of Minkowski analogs involving prescribed mean curvature and area-preserving deformations.

Analysis of G-deformations preserving principal radii and surface area.

Abstract

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.