Entire spacelike hypersurfaces of prescribed Gauss curvature in Minkowski space

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。

A Sharp Height Estimate for the Spacelike Constant Mean Curvature Graph in the Lorentz-Minkowski Space

In this paper, based on the local comparison principle in [12], we study the local behavior of the difference of two spacelike graphs in a neighborhood of a second contact point. Then we apply it to the constant mean curvature equation in 3-dimensional Lorentz-Minkowski space $\mathbb{L}^3$ and get the uniqueness of critical point for the solution of such equation over convex domain, which is an analogue of the result in [28]. Last, by this uniqueness, we obtain a minimum principle for a functional depending on the solution and its gradient. This gives us a sharp gradient estimate for the solu…

Minkowski空间中的整体类空平移孤子

Entire spacelike translating solitons in Minkowski space

本文通过在有界凸域构造凸类空解并在无穷远指定边界数据，得到了Minkowski空间中多个整体光滑凸严格类空平移孤子。

Global nonlinear stability of Minkowski space for spacelike-characteristic initial data

In this paper, we prove the global nonlinear stability of Minkowski space in the context of the spacelike-characteristic Cauchy problem for Einstein vacuum equations. Spacelike-characteristic initial data are posed on a compact 3-disk and on the future complete null hypersurface emanating from its boundary. Our result extends the seminal stability result for Minkowski space proved by Christodoulou and Klainerman for which initial data are prescribed on a spacelike 3-plane. The proof relies on the classical vectorfield method and bootstrapping argument from Christodoulou-Klainerman. The main no…

Entire spacelike radial graphs in the Minkowski space, asymptotic to the light-cone, with prescribed scalar curvature

Existence and uniqueness in ${\Bbb R}^{n,1}$ of entire spacelike hypersurfaces contained in the future of the origin $O$ and asymptotic to the light-cone, with scalar curvature prescribed at their generic point $M$ as a negative function of the unit vector $\overrightarrow{Om}$ pointing in the direction of $\overrightarrow{OM}$, divided by the square of the norm of $\overrightarrow{OM}$ (a dilation invariant problem). The solutions are seeked as graphs over the future unit-hyperboloid emanating from $O$ (the hyperbolic space); radial upper and lower solutions are constructed which, relying on …

Spacelike radial graphs of prescribed mean curvature in the Lorentz-Minkowski space

In this paper we investigate the existence and uniqueness of spacelike radial graphs of prescribed mean curvature in the Lorentz-Minkowski space $\mathbb{L}^{n+1}$, for $n\geq 2$, spanning a given boundary datum lying on the hyperbolic space $\mathbb{H}^n$.

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.

全空间样径向图在洛伦兹-闵氏空间中具有给定平均曲率的整个空间样超曲面

Entire spacelike radial graphs with prescribed mean curvature in the Lorentz--Minkowski space

研究洛伦兹-闵氏空间中具有给定平均曲率的整个空间样超曲面的存在性和唯一性，探讨其关于一点星形且渐近于光锥的性质，并推导Willmore型不等式，证明在平均曲率属于L^p时的非存在性结果。

Spacelike graphs with prescribed mean curvature on exterior domains in the Minkowski spacetime

We consider a Dirichlet problem for the mean curvature operator in the Minkowski spacetime, obtaining a necessary and sufficient condition for the existence of a spacelike solution, with prescribed mean curvature, which is the graph of a function defined on a domain equal to the complement in $\mathbb R^n$ of the union of a finite number of bounded Lipschitz domains. The mean curvature $H=H(x,t)$ is assumed to have absolute value controlled from above by a locally bounded, $L^p$-function, $p\in [1,2n/(n+2)]$, $n\geq 3$.

Spacelike convex surfaces with prescribed curvature in (2+1)-Minkowski space

We prove existence and uniqueness of solutions to the Minkowski problem in any domain of dependence $D$ in $(2+1)$-dimensional Minkowski space, provided $D$ is contained in the future cone over a point. Namely, it is possible to find a smooth convex Cauchy surface with prescribed curvature function on the image of the Gauss map. This is related to solutions of the Monge-Ampère equation $\det D^2 u(z)=(1/ψ(z))(1-|z|^2)^{-2}$ on the unit disc, with the boundary condition $u|_{\partial\mathbb{D}}=φ$, for $ψ$ a smooth positive function and $φ$ a bounded lower semicontinuous function. We then prove…

Minkowski 空间中一类曲率方程的 Dirichlet 问题

The Dirichlet problem for a class of curvature equations in Minkowski space

针对 Minkowski 空间中的一类给定曲率方程，本文通过建立先验 C² 估计，证明了一般边界数据下光滑类空超曲面解的存在性。

An invariant theory of marginally trapped surfaces in the four-dimensional Minkowski space

A marginally trapped surface in the four-dimensional Minkowski space is a spacelike surface whose mean curvature vector is lightlike at each point. We associate a geometrically determined moving frame field to such a surface and using the derivative formulas for this frame field we obtain seven invariant functions. Our main theorem states that these seven invariants determine the surface up to a motion in Minkowski space. We introduce meridian surfaces as one-parameter systems of meridians of a rotational hypersurface in the four-dimensional Minkowski space. We find all marginally trapped meri…

具有正则共形紧化的闵可夫斯基时空微扰

Perturbations of Minkowski spacetime with regular conformal compactification

本研究构造了初值逆多项式衰减至Kerr初值的闵可夫斯基时空微扰，证明其在类光、类时无穷远有正则共形紧化，推广了Friedrich的结果，核心技术是基于爱因斯坦方程新表述与定制能量估计求解类空无穷远附近的解。

Minkowski空间中预定曲率方程的Dirichlet问题

The Dirichlet problem for the prescribed curvature equations in Minkowski space

针对Minkowski空间中预定曲率类空超曲面，本文研究其Dirichlet问题，并为对应完全非线性椭圆PDE的容许解建立新的内部二阶估计。

Embeddedness of timelike maximal surfaces in (1+2) Minkowski space

We prove that if $ϕ\colon \mathbb{R}^2 \to \mathbb{R}^{1+2}$ is a smooth proper timelike immersion with vanishing mean curvature, then necessarily $ϕ$ is an embedding, and every compact subset of $ϕ(\mathbb{R}^2)$ is a smooth graph. It follows that if one evolves any smooth self-intersecting spacelike curve (or any planar spacelike curve whose unit tangent vector spans a closed semi-circle) so as to trace a timelike surface of vanishing mean curvature in $\mathbb{R}^{1+2}$, then the evolving surface will either fail to remain timelike, or it will fail to remain smooth. We show that, even allow…

Entire surfaces of constant curvature in Minkowski 3-space

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…

Minkowski空间中的平移解与全Hessian曲率流

Translating solutions and the entire Hessian curvature flow in Minkowski space

该文研究Minkowski空间非紧类空超曲面的$k$-Hessian曲率流，证明了给定渐近行为的平移解存在性，以及满足条件的严格凸初始超曲面的曲率流全局存在且归一化后收敛到平移解。

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…

Lorentz-Minkowski空间中预定平均曲率方程的精确梯度估计与$W^{2,q}$正则性

A sharp gradient estimate and $W^{2,q}$ regularity for the prescribed mean curvature equation in the Lorentz-Minkowski space

本文研究了Lorentz-Minkowski空间中预定平均曲率方程，证明了经典解的新梯度估计，并得出满足无穷远齐次边界条件的唯一弱解在特定条件下具有$W^{2,q}$正则性且严格类空。

广义 Calabi 对应与完备类空曲面

Generalized Calabi correspondence and complete spacelike surfaces

本文构造 Riemannian 与 Lorentzian Killing 浸没间给定平均曲率图的孪生对应，推广 Calabi 对应，并用于研究完备类空曲面的模空间。

The global nonlinear stability of Minkowski space for self-gravitating massive fields

The theory presented in this monograph establishes the first mathematically rigorous result on the global nonlinear stability of self-gravitating matter under small perturbations of an asymptotically flat, spacelike hypersurface of Minkowski spacetime. It allows one to exclude the existence of dynamically unstable, self-gravitating massive fields and, therefore, solves a long-standing open problem in General Relativity. By a significant extension of the Hyperboloidal Foliation Method they introduced in 2014, the authors establish global-in-time existence for the Einstein equations expressed as…

The $L_p$-Gaussian Minkowski problem

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 $p\geq1$.

The global nonlinear stability of Minkowski space for the Einstein equations in presence of a massive field

We provide a significant extension of the Hyperboloidal Foliation Method introduced by the authors in 2014 in order to establish global existence results for systems of quasilinear wave equations posed on a curved space, when wave equations and Klein-Gordon equations are coupled. This method is based on a (3+1) foliation (of the interior of a future light cone in Minkowski spacetime) by spacelike and asymptotically hyperboloidal hypersurfaces. In the new formulation of the method, we succeed to cover wave-Klein-Gordon systems containing "strong interaction" terms at the level of the metric, an…