Smooth solutions to the Gauss image problem

Abstract

In this paper we study the the Gauss image problem, which is 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.

A flow approach to the Musielak-Orlicz-Gauss image problem

“…Our approach is based on the study of suitably designed parabolic flows. The flow technique has been proved to be effective and powerful in solving the Minkowski type and Gauss image problems [7,8,13,14,15,20,37,39,40,53]. The idea behind the flow technique is the fact that the Minkowski type and Gauss image problems can be reformulated as a Monge-Ampère type equation on S n , and this indeed works for the (extended) Musielak-Orlicz-Gauss image problem due to equation (1.11) (see [31]).…”

“…As (1.12) involves the Musielak-Orlicz functions, such a flow could be named a Musielak-Orlicz-Gauss curvature flow, which is arguably the most general curvature flow related to Minkowski type and Gauss image problems and contains all previous well-studied flows [7,8,13,14,15,20,37,39,40,53] as its special cases.…”

On the Musielak-Orlicz-Gauss image problem

“…Under what conditions on λ and µ, does there exist a K ∈ K n (o) such that µ(ω) = λ(K, ω) holds for all ω ∈ B? Solutions to the Gauss image problem can be found in [8,15]. When dλ(ξ) = dξ, Problem 2.1 becomes the classical Aleksandrov problem aiming to characterize the Aleksandrov's integral curvature.…”

“…Under some mild conditions on λ and µ, the existence and uniqueness of solutions to the Gauss image problem have been established in [8]. See [15] for smooth solutions to the Gauss image problem.…”

The $L_p$ Gauss image problem

“…When λ is spherical Lebesgue measure, the L p Gauss image problem is just the L p Aleksandrov problem, see [1,2,3,25]. The L 0 Gauss image problem is just the Gauss image problem which was first mentioned in [5], and the existence of smooth solution for the Gauss image problem was in [14]. It is necessary to contrast the L p Gauss image problem with the various Minkowski problems and dual Minkowski problems that have been extensively studied, see [8,11,15,16,27,29,30,34,35,36,37,38,40,44,45] for the L p -Minkowski problem, [7,21,23,24,32,42,43] for the dual Minkowski problem, [6,9,10,25,26,31,39] for the L p dual Minkowski problem, [4,20,22,28] for the Orlicz Minkowski problem, [12,13,18,19,33] for the dual Orlicz Minkowski problem, [17] for the Orlicz Aleskandrov pr...…”