The solution of the Minkowski problem for open surfaces in Riemannian space

TL;DR

This paper addresses the Minkowski problem in Riemannian space by constructing and analyzing G-deformations that preserve the product of principal curvatures on surfaces, reducing it to a nonlinear boundary-value problem.

Contribution

It introduces a method for constructing continuous G-deformations preserving principal curvature products, extending the Minkowski problem to surfaces with boundary in Riemannian space.

Findings

G-deformations transfer normal vectors in parallel along translation paths

The equations reduce to a nonlinear boundary-value problem

Qualitative analysis of the deformation method is provided

Abstract

Author reduces the Minkowski problem to the problem of construction the G-deformations preserving the product of principal curvatures 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 continuous G-deformations preserving the product of principal curvatures of surface with boundary are considered in this article. The equations of deformations which are obtained in this paper reduce to the nonlinear boundary-value problem. The method of construction continuous G-deformations preserving the product of principal curvatures of surface with boundary and its qualitative analysis are presented in this article