Vol. 317, No. 2, 2022

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Smooth solutions to the Gauss image problem

Li Chen, Di Wu and Ni Xiang

Vol. 317 (2022), No. 2, 275–295
Abstract

We study the Gauss image problem, a generalization of the Aleksandrov problem in convex geometry. By considering a geometric flow involving Gauss curvature and functions of normal vectors and radial vectors, we obtain the existence of smooth solutions to this problem.

Keywords
Monge–Ampère equation, the Gauss image problem, Gauss curvature flow, existence of solutions
Mathematical Subject Classification 2010
Primary: 35J96
Secondary: 52A20, 53C44
Milestones
Received: 9 August 2021
Revised: 25 January 2022
Accepted: 19 February 2022
Published: 14 July 2022
Authors
Li Chen
Faculty of Mathematics and Statistics
Hubei Key Laboratory of Applied Mathematics
HuBei University
WuHan
China
Di Wu
Faculty of Mathematics and Statistics
Hubei Key Laboratory of Applied Mathematics
HuBei University
WuHan
China
Ni Xiang
Faculty of Mathematics and Statistics
Hubei Key Laboratory of Applied Mathematics
HuBei University
WuHan
China