Christoffel-Minkowski flows

双曲空间中的horospherical $p$-Christoffel-Minkowski问题

The horospherical $p$-Christoffel-Minkowski problem in hyperbolic space

本文研究双曲空间中的horospherical $p$-Christoffel-Minkowski问题，通过粘性方法证明满秩定理，建立了一致$h$-凸解的存在性，并蕴含了共形几何中Nirenberg型问题解的存在性。

无全局项的各向异性流与对偶Orlicz Christoffel-Minkowski型问题

Anisotropic flows without global terms and dual Orlicz Christoffel-Minkowski type problem

本文研究了无全局项的各向异性曲率流的长时间存在性，并利用其平稳解解决对偶Orlicz Christoffel-Minkowski型问题，涵盖多种已知问题并部分回答了一般指定曲率问题。

A class of inverse curvature flows and $L^p$ dual Christoffel-Minkowski problem

In this paper, we consider a large class of expanding flows of closed, smooth, star-shaped hypersurface in Euclidean space $\mathbb{R}^{n+1}$ with speed $ψu^αρ^δf^{-β}$, where $ψ$ is a smooth positive function on unit sphere, $u$ is the support function of the hypersurface, $ρ$ is the radial function, $f$ is a smooth, symmetric, homogenous of degree one, positive function of the principal curvatures of the hypersurface on a convex cone. When $ψ=1$, we prove that the flow exists for all time and converges to infinity if $α+δ+β\le1, β>0$ and $α\le0$, while in case $α+δ+β>1,α,δ\le0$, the flow blo…

旋转对称下Christoffel-Minkowski问题和Hessian方程的显式解

Explicit solutions to Christoffel-Minkowski problems and Hessian equations under rotational symmetries

本文在旋转对称性下，给出了凸旋转体Christoffel-Minkowski问题的显式解，并研究了混合面积测度的存在性问题，通过构造显式的凸解来解决混合Monge-Ampère方程，特别地处理了R^n上的k-Hessian方程的Dirichlet问题。

毛细Christoffel-Minkowski问题

The capillary Christoffel-Minkowski problem

本文提出毛细凸体的k-th毛细面积测度，并引入毛细Christoffel-Minkowski问题，通过求解Hessian型方程与Robin边界条件，证明了在自然充分条件下解的存在性和唯一性。

$L_{p}$ 对偶 Christoffel-Minkowski 问题（$1<p<q\leq k+1$，$1\leq k\leq n$）

The $L_{p}$ dual Christoffel-Minkowski problem for $1<p<q\leq k+1$ with $1\leq k\leq n$

本文研究 $L_p$ Christoffel-Minkowski 型问题，通过梯度估计证明了在 $1<p<q\leq k+1$ 条件下存在偶光滑严格凸解。

毛细管Christoffel-Minkowski问题

Capillary Christoffel-Minkowski problem

本文将Guan和Ma关于球面凸函数对应严格凸超曲面$σ_k$曲率的结果，推广到半空间毛细管环境中，证明当$θ\in(0,π/2)$时，满足条件的函数对应严格凸毛细管超曲面的$σ_k$曲率。

毛细管 $L_{p}$ 对偶 Christoffel-Minkowski 问题，其中 $1<p<q\leq k+1$ 且 $1\leq k\leq n$

The capillary $L_{p}$ dual Christoffel-Minkowski problem for $1<p<q\leq k+1$ with $1\leq k\leq n$

本文针对毛细管 $L_p$ 对偶 Christoffel-Minkowski 问题，通过梯度估计，在 $1<p<q\leq k+1$ 参数范围内证明了偶、光滑、严格凸解的存在性。

A unified flow approach to smooth $L^p$ Christoffel-Minkowski problem for $p>1$

In this paper we study an anisotropic expanding flow of smooth, closed, uniformly convex hypersurfaces in $\mathbb{R}^{n+1}$ with speed $ψσ_k(λ)^α$, where $α$ is a positive constant, $σ_k(λ)$ is the $k$-th elementary symmetric polynomial of the principal radii of curvature and $ψ$ is a preassigned positive smooth function defined on $\mathbb{S}^n$. We prove that under some assumptions of $ψ$, the solution to the flow after normalisation exists for all time and converges smoothly to a solution of the well-known $L^p$ Christoffel-Minkowski problem $u^{1-p}( x ) σ_k \left( \nabla^2u+uI\right)=cψ(…

Hyperbolic $p$-sum and Horospherical $p$-Brunn-Minkowski theory in hyperbolic space

The classical Brunn-Minkowski theory studies the geometry of convex bodies in Euclidean space by use of the Minkowski sum. It originated from H. Brunn's thesis in 1887 and H. Minkowski's paper in 1903. Because there is no universally acknowledged definition of the sum of two sets in hyperbolic space, there has been no Brunn-Minkowski theory in hyperbolic space since 1903. In this paper, for any $p>0$ we introduce a sum of two sets in hyperbolic space, and we call it the hyperbolic $p$-sum. Then we develop a Brunn-Minkowski theory in the hyperbolic space by use of our hyperbolic $p$-sum, and we…

Orlicz-Minkowski flows

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…

A flow method for a generalization of $L_{p}$ Christofell-Minkowski problem

In this paper, a generalization of the $L_{p}$-Christoffel-Minkowski problem is studied. We consider an anisotropic curvature flow and derive the long-time existence of the flow. Then under some initial data, we obtain the existence of smooth solutions to this problem for $c=1$.

Riemann-Christoffel flows

A geometric flow based in the Riemann-Christoffel curvature tensor that in two dimensions has some common features with the usual Ricci flow is presented. For $n$ dimensional spaces this new flow takes into account all the components of the intrinsic curvature. For four dimensional Lorentzian manifolds it is found that the solutions of the Einstein equations associated to a "detonant" sphere of matter, as well, as a Friedman-Roberson-Walker cosmological model are examples of Riemann-Christoffel flows. Possible generalizations are mentioned.

各向异性毛细超曲面的广义Minkowski公式与刚性结果

Generalized Minkowski formulas and rigidity results for anisotropic capillary hypersurfaces

该研究提出半空间中各向异性毛细超曲面的新Hsiung-Minkowski积分公式，证明相关Alexandrov型定理与刚性结果，得到各向异性Orlicz-Christoffel-Minkowski问题解的唯一性，为p≥1的L_p-Minkowski问题解的唯一性提供新证明。

A class of anisotropic expanding curvature flows

We consider an expanding flow of smooth, closed, uniformly convex hypersurfaces in (n+1)-dimensional Euclidean space with speed fu^{alpha}{sigma}_k^{beta}, where u is the support function of the hypersurface, alpha, beta are two constants, and beta>0, sigma_k is the k-th symmetric polynomial of the principle curvature radii of the hypersurface, k is an integer and 1<= k<= n. We prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalisation, to a soliton which is a solution of a elliptic equation, when the constants alpha, beta bel…

A class of curvature type equations

In this paper, we study the solvability of a general class of fully nonlinear curvature equations, which can be viewed as generalizations of the equations for Christoffel-Minkowski problem in convex geometry. We will also study the Dirichlet problem of the corresponding degenerate equations as an extension of the equations studied by Krylov.

一类非齐次曲率问题解的唯一性

Uniqueness of solutions to a class of non-homogeneous curvature problems

本文利用 Heintze-Karcher 不等式证明一类各向同性混合 Christoffel-Minkowski 及 Minkowski 型问题的光滑凸解只能是中心球面，并推出 Orlicz-Minkowski 与 $L_p$-Gaussian-Minkowski 问题的唯一性。

Hypersurfaces of Prescribed Curvature Measure

We consider the corresponding Christoffel-Minkowski problem for curvature measures. The existence of star-shaped $(n-k)$-convex bodies with prescribed $k$-th curvature measures ($k>0$) has been a longstanding problem. This is settled in this paper through the establishment of a crucial $C^2$ a priori estimate for the corresponding curvature equation on $\mathbb S^n$.

一类各向同性曲率问题解的唯一性

Uniqueness of solutions to a class of isotropic curvature problems

本文运用局部Brunn-Minkowski不等式，为Gauss曲率流闭自相似解的唯一性提供了简洁证明，并将该结论推广至多种非线性情形及偶Christoffel-Minkowski型问题。

扭曲空间中的闵可夫斯基不等式与受限逆曲率流

Minkowski inequalities and constrained inverse curvature flows in warped spaces

本文研究黎曼扭曲空间中的局部受限逆曲率流，证明了其长时间存在性与平滑收敛性，并利用单调量及逆平均曲率流推导出多维空间中新的闵可夫斯基型不等式。

Lorentzian area measures and the Christoffel problem

We introduce a particular class of unbounded closed convex sets of $\R^{d+1}$, called F-convex sets (F stands for future). To define them, we use the Minkowski bilinear form of signature $(+,...,+,-)$ instead of the usual scalar product, and we ask the Gauss map to be a surjection onto the hyperbolic space $\H^d$. Important examples are embeddings of the universal cover of so-called globally hyperbolic maximal flat Lorentzian manifolds. Basic tools are first derived, similarly to the classical study of convex bodies. For example, F-convex sets are determined by their support function, which is…

Deforming a convex hypersurface by anisotropic curvature flows

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.

A flow approach to the Orlicz-Minkowski problem for torsional rigidity

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.

A flow method for the dual Orlicz-Minkowski problem

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.

Translating solitons of mean curvature flow of noncompact spacelike hypersurfaces in Minkowski space

In this paper, we study the existence, uniqueness and asymptotic behavior of rotationally symmetric translating solitons of the mean curvature flow in Minkowski space. We also study the asymptotic behavior and the strict convexity of general solitons of such flows.

Parabolic approaches to curvature equations

We employ curvature flows without global terms to seek strictly convex, spacelike solutions of a broad class of elliptic prescribed curvature equations in the simply connected Riemannian spaceforms and the Lorentzian de Sitter space, where the prescribed function may depend on the position and the normal vector. In particular, in the Euclidean space we solve a class of prescribed curvature measure problems, intermediate $L_p$-Aleksandrov and dual Minkowski problems as well as their counterparts, namely the $L_{p}$-Christoffel-Minkowski type problems. In some cases we do not impose any conditio…

通过高斯曲率流解决p-调和测度的Minkowski问题

Flow by Gauss curvature to the Minkowski problem of p-harmonic measure

本文利用高斯曲率流方法证明了p-调和测度Minkowski问题光滑解的存在性，推广了Jerison及Akman-Mukherjee的相关结果。

A generalized Gauss curvature flow related to the Orlicz-Minkowski problem

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.

A class of anisotropic inverse Gauss curvature flows and dual Orlicz Minkowski type problem

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<…

Entire scalar curvature flow and hypersurfaces of constant scalar curvature in Minkowski space

We prove existence in the Minkowski space of entire spacelike hypersurfaces with constant negative scalar curvature and given set of lightlike directions at infinity; we also construct the entire scalar curvature flow with prescribed set of lightlike directions at infinity, and prove that the flow converges to a spacelike hypersurface with constant scalar curvature. The proofs rely on barriers construction and a priori estimates.

毛细管 $L_p$ 闵可夫斯基流

Capillary $L_p$ Minkowski Flows

本文研究各向异性毛细管高斯曲率流的长时间行为，并利用流方法证明了半空间中两类毛细管 $L_p$ 闵可夫斯基问题光滑解的存在性。

A unified flow approach to smooth, even $L_p$-Minkowski problems

We study long-time existence and asymptotic behaviour for a class of anisotropic, expanding curvature flows. For this we adapt new curvature estimates, which were developed by Guan, Ren and Wang to treat some stationary prescribed curvature problems. As an application we give a unified flow approach to the existence of smooth, even $L_p$-Minkowski problems in $\mathbb{R}^{n+1}$ for $p>-n-1.$

A flow approach to the $L_{-2}$ Minkowski problem

We prove that the set of smooth, $π$-periodic, positive functions on the unit circle for which the $L_{-2}$ Minkowski problem is solvable is dense in the set of all smooth, $π$-periodic, positive functions on the unit circle with respect to the $L^{\infty}$ norm. Furthermore, we obtain a necessary condition on the solvability of the even $ L_{-2}$ Minkowski problem. At the end, we prove uniqueness of the solutions up to an affine linear transformation.

Flow by Gauss curvature to the $L_p$-Gaussian Minkowski problem

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…

通过高斯曲率流研究 q-扭转刚度的 Orlicz Minkowski 问题

Flow by Gauss Curvature to the Orlicz Minkowski Problem for q-torsional rigidity

本文首次提出并研究 q-扭转刚度的 Orlicz Minkowski 问题，利用高斯曲率流方法证明了 q>1 时光滑非偶解的存在性，填补了该领域的研究空白。

Minkowski空间中锥体的毛细问题：关于时空平均曲率流

A capillary problem for spacelike mean curvature flow in a cone of Minkowski space

本文研究了Minkowski空间中锥体的毛细问题，证明了在特定条件下，时空平均曲率流的抛物型边界值问题存在全局解，并趋于自相似的扩张解。

Negatively Curved Three-Manifolds, Hyperbolic Metrics, Isometric Embeddings In Minkowski Space And The Cross Curvature Flow

This short note is a mostly expository article examining negatively curved three-manifolds. We look at some rigidity properties related to isometric embeddings into Minkowski space. We also review the Cross Curvature Flow (XCF) as a tool to study the space of negatively curved metrics on hyperbolic three-manifolds, the largest and least understood class of model geometries in Thurston's Geometrisation. The relationship between integrability and embedability yields interesting insights, and we show that solutions with fixed Einstein volume are precisely the integrable solutions, answering a que…

Non-contraction of heat flow on Minkowski spaces

We study contractivity properties of gradient flows for functions on normed spaces or, more generally, on Finsler manifolds. Contractivity of the flows turns out to be equivalent to a new notion of convexity for the functions. This is different from the usual convexity along geodesics in non-Riemannian Finsler manifolds. As an application, we show that the heat flow on Minkowski normed spaces other than inner product spaces is not contractive with respect to the quadratic Wasserstein distance.

Entire $σ_k$ curvature flow in Minkowski space

In this paper, we study the $σ_k$ curvature flow of noncompact spacelike hypersurfaces in Minkowski space. We prove that if the initial hypersurface satisfies certain conditions, then the flow exists for all time. Moreover, we show that after rescaling, the flow converges to a self-expander.

Entire convex curvature flow in Minkowski space

In this paper, we study fully nonlinear curvature flows of noncompact spacelike hypersurfaces in Minkowski space. We prove that if the initial hypersurface satisfies certain conditions, then the flow exists for all time. Moreover, we show that after rescaling the flow converges to the future timelike hyperboloid, which is a self-expander.

闵可夫斯基空间中的逆Hessian曲率流 I: 协紧超曲面

Inverse Hessian Curvature Flow in Minkowski Space I: Cocompact Hypersurfaces

研究闵可夫斯基空间中协紧逆σ_k曲率流，证明其长时间存在性和收敛性，并建立quermassintegral不等式。

Self-similar solutions to the mean curvature flow in the Minkowski plane $\mathbf R^{1,1}$

We introduce the mean curvature flow of curves in the Minkowski plane $\mathbf R^{1,1}$ and give a classification of all the self-similar solutions. In addition, we describe five other exact solutions to the flow.

Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems

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…

通过高斯曲率流求解 Orlicz 弦 Minkowski 问题

Flow by Gauss Curvature to the orlicz Chord Minkowski Problem

本文研究 orlicz 弦 Minkowski 问题，通过用连续函数 $\varphi$ 替换 $p$ 推广了 $L_p$ 弦 Minkowski 问题，并利用高斯曲率流方法证明了其光滑解的存在性。

The Mean Curvature Flow in Minkowski Spaces

Studying the geometric flow plays a powerful role in mathematics and physics. In this paper, we introduce the mean curvature flow on Finsler manifolds and give a number of examples of the mean curvature flow. For Minkowski spaces, a special case of Finsler manifolds, we will prove the existence and uniqueness for solution of the mean curvature flow and prove that the flow preserves the convexity and mean convexity. We also derive some comparison principles for the mean curvature flow.

上半空间中的毛细Orlicz-闵可夫斯基流

Capillary Orlicz-Minkowski flow in the upper half-space

本文研究了上半空间中各向异性毛细高斯曲率流的长时间存在性和渐进行为，通过流方法解决了毛细Orlicz-闵可夫斯基问题的存在性问题，无需偶性假设。

An Anisotropic shrinking flow and L_p Minkowski problem

We consider a shrinking flow of smooth, closed, uniformly convex hypersurfaces in (n+1)-dimensional Euclidean space with speed fu^{alpha}{sigma}_n^{beta}, where u is the support function of the hypersurface, alpha, beta are two constants, and beta>0, sigma_n is the n-th symmetric polynomial of the principle curvature radii of the hypersurface. We prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalisation, to a soliton which is a solution of an elliptic equation, when the constants alpha, beta belong to a suitable range, provid…

关于闵可夫斯基空间中平均曲率流的第二边值问题

On the second boundary value problem for mean curvature flow in Minkowski space

本文作为先前工作的续篇，在闵可夫斯基空间中构造了具有指定高斯像的平均曲率流平移孤立子，推进了第二边值问题的研究。

Inverse anisotropic mean curvature flow and a Minkowski type inequality

In this paper, we show that the inverse anisotropic mean curvature flow in $\mathbb{R}^{n+1}$, initiating from a star-shaped, strictly $F$-mean convex hypersurface, exists for all time and after rescaling the flow converges exponentially fast to a rescaled Wulff shape in the $C^\infty$ topology. As an application, we prove a Minkowski type inequality for star-shaped, $F$-mean convex hypersurfaces.

Ricci flow on surfaces along the standard lightcone in the $3+1$-Minkowski spacetime

Identifying any conformally round metric on the $2$-sphere with a unique cross section on the standard lightcone in the $3+1$-Minkowski spacetime, we gain a new perspective on $2d$-Ricci flow on topological spheres. It turns out that in this setting, Ricci flow is equivalent to a null mean curvature flow first studied by Roesch--Scheuer along null hypersurfaces. Exploiting this equivalence, we can translate well-known results from $2d$-Ricci flow first proven by Hamilton into a full classification of the singularity models for null mean curvature flow in the Minkowski lightcone. Conversely, we…

On the Minkowski-type inequality for outward minimizing hypersurfaces in Schwarzschild space

Using the weak solution of Inverse mean curvature flow, we prove the sharp Minkowski-type inequality for outward minimizing hypersurfaces in Schwarzschild space.

Cartan-Hadamard 流形中的 Minkowski 不等式

Minkowski inequality in Cartan-Hadamard manifolds

本文利用调和平均曲率流证明 Cartan-Hadamard 三维流形中给定面积凸曲面的精确总平均曲率下界，并推出非正曲率空间中的 Bonnesen 型等周不等式，且将联系推广至任意维数。

Minkowski Inequalities via Nonlinear Potential Theory

In this paper, we prove an extended version of the Minkowski Inequality, holding for any smooth bounded set $Ω\subset \mathbb R^n$, $n\geq 3$. Our proof relies on the discovery of effective monotonicity formulas holding along the level set flow of the $p$-capacitary potentials associated with $Ω$, for every $p$ sufficiently close to $1$. Besides constituting a neat improvement of those introduced in [Fog_Maz_Pin] to treat the case of convex domains, these formulas testify the existence of a link between the monotonicity formulas derived by Colding and Minicozzi for the level set flow of Green'…

Entire self-expanders for power of $σ_k$ curvature flow in Minkowski space

In [19], we prove that if an entire, spacelike, convex hypersurface $\mathcal{M}_{u_0}$ has bounded principal curvatures, then the $σ_k^{1/α}$ (power of $σ_k$) curvature flow starting from $\mathcal{M}_{u_0}$ admits a smooth convex solution $u$ for $t>0.$ Moreover, after rescaling, the flow converges to a convex self-expander $\tilde{\mathcal{M}}=\{(x, \tilde{u}(x))\mid x\in\mathbb{R}^n\}$ that satisfies $σ_k(κ[\tilde{\mathcal{M}}])=(-\left<X_0, ν_0\right>)^α.$ Unfortunately, the existence of self-expander for power of $σ_k$ curvature flow in Minkowski space has not been studied before. In thi…