Rafael López
In Lorentz-Minkowski space, we prove that the conjugate surface of a maximal graph over a convex domain is also a graph. We provide three proofs of this result that show a suitable correspondence between maximal surfaces in Lorentz-Minkowski space and minimal surfaces in Euclidean space.
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Several characterizations of umbilic points of submanifolds in arbitrary Riemannian and Lorentzian manifolds are given. As a consequence, we obtain new characterizations of spheres in the Euclidean space and of hyperbolic spaces in the Lorentz-Minkowski space. We also prove the Lorentzian version of a classical result by Cartan.
We prove that the conformal group of a closed, simply connected, real analytic Lorentzian manifold is compact. D'Ambra proved in 1988 that the isometry group of such a manifold is compact. Our result implies the Lorentzian Lichnerowicz Conjecture for real analytic Lorentzian manifolds with finite fundamental group. Third version includes corrections and clarifications, particularly in section 6.
We introduce a version of Aubry-Mather theory for the length functional of causal curves in compact Lorentzian manifolds. Results include the existence of maximal invariant measures, calibrations and calibrated curves. We prove two versions of the Mather's graph theorem. A class of examples, the Lorentzian Hedlund examples, shows the optimality of the obtained results.
Riemannian and Lorentzian Calderón problem under Magnetic Perturbation
本文研究了磁扰动下黎曼和洛伦兹卡德龙问题的唯一确定性及共形类构造。
A note on the Lorentzian splitting theorem
本文在弱化里奇曲率条件下证明洛伦兹分裂定理，利用非时性极限与几何极大值原理，去除了全局双曲情形中的有界性条件，强化了已有结果。
In this paper we establish some parabolicity criteria for maximal surfaces immersed into a Lorentzian product space of the form $M^2\times\mathbb{R}_1$, where $M^2$ is a connected Riemannian surface with non-negative Gaussian curvature and $M^2\times\mathbb{R}_1$ is endowed with the Lorentzian product metric $<,>=<,>_M-dt^2$. In particular, and as an application of our main result, we deduce that every maximal graph over a starlike domain $Ω\subseteq M$ is parabolic. This allows us to give an alternative proof of the non-parametric version of the Calabi-Bernstein result for entire maximal grap…
In this article we introduce a notion of normalized angle for Lorentzian pre-length spaces. This concept allows us to prove some equivalences to the definition of timelike curvature bounds from below for Lorentzian pre-length spaces. Specifically, we establish some comparison theorems known as the local Lorentzian version of the Toponogov theorem and the Alexandrov convexity property. Finally, as an application we obtain a first variation Formula for non-negatively curved globally hyperbolic Lorentzian length spaces.
Let S be a compact surface of genus >1, and g be a metric on S of constant curvature K\in\{-1,0,1\} with conical singularities of negative singular curvature. When K=1 we add the condition that the lengths of the contractible geodesics are >2π. We prove that there exists a convex polyhedral surface P in the Lorentzian space-form of curvature K and a group G of isometries of this space such that the induced metric on the quotient P/G is isometric to (S,g). Moreover, the pair (P,G) is unique (up to global isometries) among a particular class of convex polyhedra, namely Fuchsian polyhedra. This e…
Lorentzian manifolds properly isometrically embeddable in Minkowski spacetime
该研究刻画了可恰当等距嵌入闵可夫斯基时空的洛伦兹流形，证明可恰当共形嵌入的这类流形等价于全局双曲时空，并借助嵌入得到距离公式的无穷小版本。
The c-completion of Lorentzian metric spaces
本文重新审视洛伦兹度量空间概念，在极大普遍性下构建了包含低正则性时空的c-完备化，并证明其自身具有洛伦兹度量空间结构且能精确完备原空间并反映其因果性质。
A Toponogov globalisation result for Lorentzian length spaces
本文在合成洛伦兹长度空间中建立 Toponogov 全局化定理，利用猫摇篮构造及三角形细分和 Lebesgue 数引理，并应用于 Bonnet–Myers 定理、分裂定理及曲率界稳定性。
Spacetime closedness theorem for homogeneous Lorentzian foliations
该文提出齐次洛伦兹叶状结构的时空闭包定理，通过分析步骤从大爆炸和时间控制得有限切片体积，再经几何步骤提升为紧致性，并给出Lean 4形式化证明。
A local Lorentzian Ferrand-Obata theorem for conformal vector fields
研究闭实解析洛伦兹流形上共形向量场，通过改进局部范式等方法，证明其流局部等距或度量处处共形平坦，此为洛伦兹 - 利希纳罗维茨猜想的局部版本。
Lorentzian versions of classical Riemannian volume comparison theorems by Gunther, Bishop and Bishop-Gromov, are stated for suitable natural subsets of general semi-Riemannian manifolds. The problem is more subtle in the Bishop-Gromov case, which is extensively discussed. For the general semi-Riemannian case, a local version of the Gunther and Bishop theorems is given and applied.
A Lorentzian Equivariant Index Theorem
本文提出一种洛伦兹时空下扭曲狄拉克算子的等价指标公式，结合等价和非等价情况的简单技术，推导出洛伦兹“指标=谱流”公式的等价形式。
Semiglobal uniqueness for the Lorentzian Calderón problem
该研究在半全局框架下证明洛伦兹 Calderón 问题的唯一性，采用畸变平面波与加权 $L^2$ 估计，无需控制论论证等传统方法，仅用更少数据求解问题的形式确定版本。
Low regularity approach to Bartnik's conjecture
本文研究洛伦兹长度空间下的Bartnik分裂猜想，通过证明因果边界单点性，得到满足类时完备、非负类时曲率界的紧底空间乘积型整体双曲洛伦兹长度空间的度量分裂结果。
Motivated by the classical Euler elastic curves, David A. Singer posed in 1999 the problem of determining a plane curve whose curvature is given in terms of its position. We propound the same question in Lorentz-Minkowski plane, focusing on spacelike and timelike curves. In this article, we study those curves in $\mathbb{L}^2$ whose curvature depends on the Lorentzian pseudodistance from the origin, and those ones whose curvature depends on the Lorentzian pseudodistance through the horizontal or vertical geodesic to a fixed lightlike geodesic. Making use of the notions of geometric angular mom…
We prove several geometric theorems using tools from the theory of convex optimization. In the Riemannian setting, we prove the max flow-min cut theorem for boundary regions, applied recently to develop a "bit-thread" interpretation of holographic entanglement entropies. We also prove various properties of the max flow and min cut, including respective nesting properties. In the Lorentzian setting, we prove the analogous min flow-max cut theorem, which states that the volume of a maximal slice equals the flux of a minimal flow, where a flow is defined as a divergenceless timelike vector field …
Killing Vectors and Bochner-Type Theorems in Pseudo-Riemannian Hom-Lie Algebras
研究伪黎曼Hom-李代数，建立黎曼和洛伦兹情形下经典Bochner定理的Hom-版本，并证明Killing向量空间构成全测地Hom-李子代数。
Quantitative Lorentzian isoperimetric inequalities in conical Minkowski spacetimes
研究在锥形闵可夫斯基时空中建立洛伦兹等周不等式的稳定性估计，通过Fraenkel不对称性，针对不同不等式有不同形式，还可改进并给出简单证明，利用因果结构的利普希茨界提升为豪斯多夫稳定性估计。
In this paper we establish new Calabi-Bernstein results for maximal surfaces immersed into a Lorentzian product space of the form $M^2\times\mathbb{R}_1$, where $M^2$ is a connected Riemannian surface and $M^2\times\mathbb{R}_1$ is endowed with the Lorentzian metric $<,>=<,>_{M}-dt^2$. In particular, when $M$ is a Riemannian surface with non-negative Gaussian curvature $K_M$, we prove that any complete maximal surface in $M^2\times\mathbb{R}_1$ must be totally geodesic. Besides, if $M$ is non-flat we conclude that it must be a slice $M\times\{t_0\}$, $t_0\in\mathbb{R}$ (here by "complete" it i…
The Hawking-Penrose singularity theorem for $C^{1,1}$-Lorentzian metrics
本文证明 Hawking-Penrose 奇点定理及其推广在 $C^{1,1}$-Lorentzian 度量下依然成立，通过提出弱版本的能量条件与一般性条件，并利用正则化方法分析矩阵 Riccati 方程，证明因果测地线必然变为非最大化的。
The Hawking Singularity Theorem for Hölder Continuous Metrics with $L^p$-Bounded Curvature
本文在低正则性条件下证明了霍金奇点定理，研究了时空度量的正则性和曲率限制，扩展了之前的结果并证明了因果平坦性和Myers定理的低正则性版本。
The harmonic oscillator as a distinguished dynamical system can be defined not only on the Euclidean plane but also on the sphere and on the hyperbolic plane, and more generally on any configuration space with constant curvature and with a metric of any signature, either Riemannian (definite positive) or Lorentzian (indefinite). In this paper we study the main properties of these `curved' harmonic oscillators simultaneously on any such configuration space, using a Cayley-Klein (CK) type approach, with two free parameters $\ki, \kii$ which altogether correspond to the possible values for curvat…
We formulate and prove the Lorentzian version of the positive mass theorems with arbitrary negative cosmological constant for asymptotically AdS spacetimes. This work is the continuation of the second author's recent work on the positive mass theorem on asymptotically hyperbolic 3-manifolds.
Rényi's entropy on Lorentzian spaces. Timelike curvature-dimension conditions
本研究在带测度的洛伦兹空间中引入基于Rényi熵凸性的类时曲率-维数条件与测度收缩性质，证明其相容性、稳定性、强弱等价等性质，并明确类时p-本质非分支的弱条件作用。
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