In this paper we study a normalised anisotropic Gauss curvature flow of strictly convex, closed hypersurfaces in the Euclidean space R^n+1. We prove that the flow exists for all time and converges smoothly to the unique, strictly convex solution of a Monge-Amp`ere type equation. Our argument provides a parabolic proof in the smooth category for the existence of solutions to the Dual Orlicz-Minkowski problem introduced by Zhu, Xing and Ye.
In this paper, combining the $p$-capacity for $p\in (1, n)$ with the Orlicz addition of convex domains, we develop the $p$-capacitary Orlicz-Brunn-Minkowski theory. In particular, the Orlicz $L_ϕ$ mixed $p$-capacity of two convex domains is introduced and its geometric interpretation is obtained by the $p$-capacitary Orlicz-Hadamard variational formula. The $p$-capacitary Orlicz-Brunn-Minkowski and Orlicz-Minkowski inequalities are established, and the equivalence of these two inequalities are discussed as well. The $p$-capacitary Orlicz-Minkowski problem is proposed and solved under some mild…
In this paper the Orlicz-Minkowski problem for torsional rigidity, a generalization of the classical Minkowski problem, is studied. Using the flow method, we obtain a new existence result of solutions to this problem for general measures.
In this paper the dual Orlicz-Minkowski problem, a generalization of the $L_p$ dual Minkowski problem, is studied. By studying a flow involving the Gauss curvature and support function, we obtain a new existence result of solutions to this problem for smooth measures.
In this paper we study the dual Orlicz-Minkowski problem, which is a generalization of the dual Minkowski problem in convex geometry. By considering a geometric flow involving Gauss curvature and functions of normal vectors and radial vectors, we obtain a new existence result of smooth even solutions to this problem for smooth even measures.
In this paper a generalized Gauss curvature flow about a convex hypersurface in the Euclidean $n$-space is studied. This flow is closely related to the Orlicz-Minkowski problem, which involves Gauss curvature and a function of support function. Under some appropriate assumptions, we prove the long-time existence and convergence of this flow. As a byproduct, two existence results of solutions to the even Orlicz-Minkowski problem are obtained, one of which improves the known result.
In this paper, we study the long-time existence and asymptotic behavior for a class of anisotropic inverse Gauss curvature flows. By the stationary solutions of anisotropic flows, we obtain some new existence results for the dual Orlicz Minkowski type problem and even dual Orlicz Minkowski type problem for smooth measures, which is the most reasonable extension of the $L^p$ dual Minkowski problem from the dual point of view. The results of corresponding $L^p$ versions are $L^p$ dual Minkowski problem for $p>q$; and even $L^p$ dual Minkowski problem for $p>-1$, or $q<1$, or some ranges of $p<0<…
We study the long-time existence and behavior for a class of anisotropic non-homogeneous Gauss curvature flows whose stationary solutions, if exist, solve the regular Orlicz-Minkowski problems. As an application, we obtain old and new results for the regular even Orlicz-Minkowski problems; the corresponding $L_p$ version is the even $L_p$-Minkowski problem for $p>-n-1$. Moreover, employing a parabolic approximation method, we give new proofs of some of the existence results for the general Orlicz-Minkowski problems; the $L_p$ versions are the even $L_p$-Minkowski problem for $p>0$ and the $L_p…
The general volume of a star body, a notion that includes the usual volume, the $q$th dual volumes, and many previous types of dual mixed volumes, is introduced. A corresponding new general dual Orlicz curvature measure is defined that specializes to the $(p,q)$-dual curvature measures introduced recently by Lutwak, Yang, and Zhang. General variational formulas are established for the general volume of two types of Orlicz linear combinations. One of these is applied to the Minkowski problem for the new general dual Orlicz curvature measure, giving in particular a solution to the Minkowski prob…
For $K\subseteq \mathbb{R}^n$ a convex body with the origin $o$ in its interior, and $ϕ:\mathbb{R}^n\setminus\{o\}\rightarrow(0, \infty)$ a continuous function, define the general dual ($L_ϕ)$ Orlicz quermassintegral of $K$ by $$\mathcal{V}_ϕ(K)=\int_{\mathbb{R}^n \setminus K} ϕ(x)\,dx.$$ Under certain conditions on $ϕ$, we prove a variational formula for the general dual ($L_ϕ)$ Orlicz quermassintegral, which motivates the definition of $\widetilde{C}_{ϕ,\mathcal{V}}(K, \cdot)$, the general dual ($L_ϕ)$ Orlicz curvature measure of $K$. We pose the following general dual Orlicz-Minkowski probl…
The general dual volume $\dveV(K)$ and the general dual Orlicz curvature measure $\deV(K, \cdot)$ were recently introduced for functions $G: (0, \infty)\times \sphere\rightarrow (0, \infty)$ and convex bodies $K$ in $\R^n$ containing the origin in their interiors. We extend $\dveV(K)$ and $\deV(K, \cdot)$ to more general functions $G: [0, \infty)\times \sphere\rightarrow [0, \infty)$ and to compact convex sets $K$ containing the origin (but not necessarily in their interiors). Some basic properties of the general dual volume and of the dual Orlicz curvature measure, such as the continuous depe…
In this paper, the extended Musielak-Orlicz-Gauss image problem is studied. Such a problem aims to characterize the Musielak-Orlicz-Gauss image measure $\widetilde{C}_{G,Ψ,λ}(Ω,\cdot)$ of convex body $Ω$ in $\mathbb{R}^{n+1}$ containing the origin (but the origin is not necessary in its interior). In particular, we provide solutions to the extended Musielak-Orlicz-Gauss image problem based on the study of suitably designed parabolic flows, and by the use of approximation technique (for general measures). Our parabolic flows involve two Musielak-Orlicz functions and hence contain many well-stud…
In the present paper we initiate the study of the Musielak-Orlicz-Brunn-Minkowski theory for convex bodies. In particular, we develop the Musielak-Orlicz-Gauss image problem aiming to characterize the Musielak-Orlicz-Gauss image measure of convex bodies. For a convex body $K$, its Musielak-Orlicz-Gauss image measure, denoted by $\widetilde{C}_Θ(K, \cdot)$, involves a triple $Θ=(G, Ψ, λ)$ where $G$ and $Ψ$ are two Musielak-Orlicz functions defined on $S^{n-1}\times (0, \infty)$ and $λ$ is a nonzero finite Lebesgue measure on the unit sphere $S^{n-1}$. Such a measure can be produced by a variati…
Existence of symmetric solutions to the Gaussian Minkowski problem was established by Huang, Xi and Zhao. In this paper, we show the existence of non-symmetric solutions to this problem by studying the related Monge-Ampère type equation on the sphere.
In this paper, we extend the article that Minkowski problem in Gaussian probability space of Huang et al. to $L_p$-Gaussian Minkowski problem, and obtain the existence and uniqueness of $o$-symmetry weak solution in case of $p\geq1$.
We prove existence and stability of smooth entire strictly convex spacelike hypersurfaces of prescribed Gauss curvature in Minkowski space. The proof is based on barrier constructions and local a priori estimates.
In this paper, we study the $L_p$-Gaussian Minkowski problem, which arises in the $L_p$-Brunn-Minkowski theory in Gaussian probability space. We use Aleksandrov's variational method with Lagrange multipliers to prove the existence of the logarithmic Gauss Minkowski problem. We construct a suitable Gauss curvature flow of closed, convex hypersurfaces in the Euclidean space $\mathbb{R}^{n+1}$, and prove its long-time existence and converges smoothly to a smooth solution of the normalized $L_p$ Gaussian Minkowski problem in cases of $p>0$ and $-n-1<p\leq 0$ with even prescribed function respectiv…
The current work focuses on the Gaussian-Minkowski problem in dimension 2. In particular, we show that if the Gaussian surface area measure is proportional to the spherical Lebesgue measure, then the corresponding convex body has to be a centered disk. As an application, this ``uniqueness'' result is used to prove the existence of smooth small solutions to the Gaussian-Minkowski problem via a degree-theoretic approach.
We are concerned with spacelike convex hypersurfaces of positive constant (K-hypersurfaces) or prescribed Gauss curvature in Minkowski space. Our main purpose is to study entire solutions as well as the Dirichlet problem in bounded domains of the related Monge-Ampere equation.
Existence of symmetric (resp. asymmetric) solutions to the $L_p$ Gaussian Minkowski problem for $p\leq 0$ (resp. $p\geq 1$) will be provided. Moreover, existence and uniqueness of smooth solutions to the problem for $p>n$ will also be proved without the restriction that the Gaussian volumes of convex bodies are not less than one-second.
In this paper, we prove the uniqueness of solutions to the logarithmic Minkowski problem in $\mathbb{R}^3$ without symmetry condition, provided the density of the measure is close to $1$ in $C^α$ norm. This result also implies the uniqueness of self-similar solutions to the anisotropic Gauss curvature flow in $\mathbb{R}^3$ when the speed function is $C^α$ close to a positive constant.
In this paper, we consider a fully nonlinear curvature flow of a convex hypersurface in the Euclidean n-space. This flow involves k-th elementary symmetric function for principal curvature radii and a function of support function. Under some appropriate assumptions, we prove the long-time existence and convergence of this flow. As an application, we give the existence of smooth solutions to the Orlicz-Christoffel-Minkowski problem.
In this paper we study a contracting flow of closed, convex hypersurfaces in the Euclidean space $\mathbb R^{n+1}$ with speed $f r^α K$, where $K$ is the Gauss curvature, $r$ is the distance from the hypersurface to the origin, and $f$ is a positive and smooth function. If $α\ge n+1$, we prove that the flow exists for all time and converges smoothly after normalisation to a soliton, which is a sphere centred at the origin if $f \equiv 1$. Our argument provides a parabolic proof in the smooth category for the classical Aleksandrov problem, and resolves the dual q-Minkowski problem introduced by…
This paper concerns the global theory of properly embedded spacelike surfaces in three-dimensional Minkowski space in relation to their Gaussian curvature. We prove that every regular domain which is not a wedge is uniquely foliated by properly embedded convex surfaces of constant Gaussian curvature. This is a consequence of our classification of surfaces with bounded prescribed Gaussian curvature, sometimes called the Minkowski problem, for which partial results were obtained by Li, Guan-Jian-Schoen, and Bonsante-Seppi. Some applications to minimal Lagrangian self-maps of the hyperbolic plane…
In this paper, we prove the existence of smooth, entire, strictly convex, spacelike, constant $σ_k$ curvature hypersurfaces with prescribed lightlike directions in Minkowski space. This is equivalent to prove the existence of smooth, entire, strictly convex, spacelike, constant $σ_k$ curvature hypersurfaces with prescribed Gauss map image. We also show that there doesn't exist any entire, convex, strictly spacelike, constant $σ_k$ curvature hypersurfaces. Moreover, we generalize the result in \cite{RWX} and construct strictly convex, spacelike, constant $σ_k$ curvature hypersurface with bounde…
In this paper, we study the anisotropic Minkowski problem. It is a problem of prescribing the anisotropic Gauss-Kronecker curvature for a closed strongly convex hypersurface in Euclidean space as a function on its anisotropic normals in relative or Minkowski geometry. We first formulate such problem to a Monge-Ampére type equation on the anisotropic support function and then prove the existence and uniqueness of the admissible solution to such equation. In conclusion, we give an affirmative answer to the anisotropic Minkowski problem.
