Chung-Jun Tsai, Kai-Hsiang Wang
We prove an isoperimetric-type inequality for maximal, spacelike submanifold in the Minkowski space. The argument is based on the recent work of Brendle.
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We obtain an estimate of the Cheeger isoperimetric constant in terms of the volume growth for a properly immersed submanifold in a Riemannian manifold which possesses at least one pole and sectional curvature bounded from above .
Based on Markvorsen and Palmer's work on mean time exit and isoperimetric inequalities we establish slightly better isoperimetric inequalities and mean time exit estimates for minimal submanifolds of $N\times\mathbb{R}$. We also prove isoperimetric inequalities for submanifolds of Hadamard spaces with tamed second fundamental form.
The aim of this paper is to prove isoperimetric inequalities on submanifolds of the Euclidean space using mass transportation methods. We obtain a sharp ?weighted isoperimetric inequality? and a nonsharp classical inequality similar to the one obtained by J. Michael and L. Simon. The proof relies on the description of a solution of the problem of Monge when the initial measure is supported in a submanifold and the final one supported in a linear subspace of the same dimension.
In the present note, first we derive an intrinsic inequality for Pseudo-umbilical spacelike submanifold in an indefinite space form. We use this inequality to show that such submanifold is totally geodesic. In the rest part of this paper, using a result of Aiyama [1], we prove that Pseudo-umbilical spacelike subamnifold is totally umbilical.
In this paper we first prove some linear isoperimetric inequalities for submanifolds in the de Sitter-Schwarzschild and Reissner-Nordstrom manifolds. Moreover, the equality is attained. Next, we prove some monotonicity formulas for submanifolds with bounded mean curvature vector in warped product manifolds and, as consequences, we give lower bound estimates for the volume of these submanifolds in terms of the warping function. We conclude the paper with an isoperimetric inequality for minimal surfaces.
We give simple conditions on an ambient manifold that are necessary and sufficient for isoperimetric inequalities (for submanifolds) to hold.
New rigidity results for complete non-compact spacelike submanifolds of arbitrary codimension in plane fronted waves are obtained. Under appropriate assumptions, we prove that a complete spacelike submanifold in these spacetimes is contained in a characteristic lightlike hypersurface. Moreover, for a complete codimension two extremal submanifold in a plane fronted wave we show sufficient conditions to guarantee that it is a (totally geodesic) wavefront.
We use a locally constrained mean curvature flow to prove the isoperimetric inequality for spacelike domains in generalized Robertson-Walker spaces satisfying the null convergence condition.
In this paper we provide an extension to the Jellett-Minkowski's formula for immersed submanifolds into ambient manifolds which possesses a pole and radial curvatures bounded from above or below by the radial sectional curvatures of a rotationally symmetric model space. Using this Jellett-Minkowski's generalized formula we can focus on several isoperimetric problems. More precisely, on lower bounds for isoperimetric quotients of any precompact domain with smooth boundary, or on the isoperimetric profile of the submanifold and its modified volume. In the particular case of a model space with st…
We introduce the totally absolute lightcone curvature for a spacelike submanifold with general codimension and investigate global properties of this curvature. One of the consequences is that the Chern-Lashof type inequality holds. Then the notion of lightlike tightness is naturally induced.
Anti-de Sitter space is the Lorentzian space form with negative curvature. In this paper we consider lightlike hypersurfaces along spacelike submanifolds in anti-de Sitter space with general codimension. In particular, we investigate the singularities of lightlike hypersurfaces as an application of the theory of Legendrian singularities.
High codimension mean curvature flow of spacelike-convex submanifolds with one spacelike codimension
研究伪欧氏空间中高码维类空子流形的平均曲率流，证明初始紧致类空凸子流形在流下曲率收缩和非坍缩量保持，并证明Huisken和Gage-Hamilton定理的类比结果。
A new approach to the study of spacelike submanifolds in a spherical Robertson-Walker spacetime: characterization of the stationary spacelike submanifolds as an application
本文利用球面 RW 时空到 Lorentz-Minkowski 时空的自然余一维等距嵌入，将其中的类空子流形置于高维背景中研究，并推广 Takahashi 定理以刻画稳态类空子流形。
In this paper we study weakly irreducible holonomy representations of the normal connection of a spacelike submanifold in a pseudo-Riemannian space from. We associate screen representations to weakly irreducible normal holonomy groups and classify the screen representations having the Borel-Lichnérowicz property. In particular, we derive a classification of Lorentzian normal holonomy representations.
For every codimension two spacelike submanifold of a Lorentz manifold and each choice of a normal lightlike vector field, we introduce a canonical way to construct a tractor conformal bundle. We characterize when the induced connection of a such submanifold defines a tractor connection and then, in this case, when this tractor conformal bundle with the induced connection is standard for the induced metric. Finally, the normality condition for this tractor conformal bundle endowed with the induced connection is characterized in terms of a strong relationship between the intrinsic and the extrin…
By means of a family of counter-examples, it is shown that the Reilly upper bound for the first eigenvalue of the Laplace operator for a compact submanifold in Euclidean space does not work for $n$-dimensional compact spacelike submanifolds of Lorentz-Minkowski spacetime $\mathbb{L}^m$, $m\geq n+2$. We develop a new suitable technique, based on an integral formula on compact spacelike sections of the light cone in $\mathbb{L}^m$. Then, a family of extrinsic upper bounds for the first eigenvalue of the Laplace operator for a compact spacelike submanifold in $\mathbb{L}^m$ is proved. For each on…
On the scalar curvature of complete maximal spacelike submanifolds in pseudo-hypebolic spaces
本文研究了伪双曲空间中完备极大类空子流形的曲率，证明了其标量曲率在任意符号下均非正并给出精确界限，显式刻画了达到界限的子流形，并在余维数为1时推导出Ricci曲率的精确上界。
Spacelike Submanifolds of Codimension Two with Parallel Mean Curvature Vector Field in Lorentz-Minkowski Spacetime Contained in the Light Cone
该研究针对洛伦兹-闵可夫斯基时空中余维二、具平行平均曲率向量场的紧致类空子流形建立积分不等式，推导出刚性结论，完整刻画了光锥内这类子流形必为类空超平面中的全脐球面。
Codimension two spacelike submanifolds into null hypersurfaces of generalized Robertson-Walker spacetimes
本文研究广义Robertson-Walker时空中类光锥内的余维二类空子流形，给出径向坐标条件使其分别共形微分同胚于双曲空间、圆圆柱面或球面，并得到弱俘获子流形的非存在性结果。
Sobolev Inequalities in Spacelike Submanifolds of Minkowski Space
本文将ABP估计方法应用于Minkowski空间$\mathbb R^{n,1}$中的类空子流形，导出了Michael-Simon型不等式，并在超曲面平均凸的条件下得到了不含平均曲率项的Sobolev不等式。
In this paper we prove general inequalities involving the weighted mean curvature of compact submanifolds immersed in weighted manifolds. As a consequence we obtain a relative linear isoperimetric inequality for such submanifolds. We also prove an extrinsic upper bound to the first non zero eigenvalue of the drift Laplacian on closed submanifolds of weighted manifolds.
In this paper we obtain an existence theorem for normal geodesics joining two given submanifolds in a globally hyperbolic stationary spacetime. The proof is based on both variational and geometric arguments involving the causal structure of the spacetime, the completeness of suitable Finsler metrics associated to it and some basic properties of a submersion. By this interaction, unlike previous results on the topic, also non--spacelike submanifolds can be handled.
Let $Σ$ be a $k$-dimensional complete proper minimal submanifold in the Poincaré ball model $B^n$ of hyperbolic geometry. If we consider $Σ$ as a subset of the unit ball $B^n$ in Euclidean space, we can measure the Euclidean volumes of the given minimal submanifold $Σ$ and the ideal boundary $\partial_\infty Σ$, say $\rvol(Σ)$ and $\rvol(\partial_\infty Σ)$, respectively. Using this concept, we prove an optimal linear isoperimetric inequality. We also prove that if $\rvol(\partial_\infty Σ) \geq \rvol(\mathbb{S}^{k-1})$, then $Σ$ satisfies the classical isoperimetric inequality. By proving the…
In this paper, we prove a Sobolev and isoperimetric inequalities for submanifold in weighted manifold. Our results generalize the Hoffman-Spruck's inequalities.
Codimension two spacelike submanifolds in Lorentzian manifolds and conformal structures
本文提出通过黎曼共形结构和单参数自伴张量场族构造洛伦兹流形，使共形类中的度量对应余维二类空子流形的诱导度量，并建立其类光法向量场的 Weingarten 自同态与 Möbius 结构之间的双向对应关系，进而用外蕴几何刻画平坦 Möbius 结构。
An isoperimetric type inequality in De Sitter space
本文证明了de Sitter空间中类空、紧致、星形且$2$-凸超曲面的最优等周不等式。
We prove a Sobolev inequality which holds on submanifolds in Euclidean space of arbitrary dimension and codimension. This inequality is sharp if the codimension is at most 2. As a special case, we obtain a sharp isoperimetric inequality for minimal submanifolds in Euclidean space of codimension at most 2.
For compact Riemannian manifolds with convex boundary, B.White proved the following alternative: Either there is an isoperimetric inequality for minimal hypersurfaces or there exists a closed minimal hypersurface, possibly with a small singular set. There is the natural question if a similar result is true for submanifolds of higher codimension. Specifically, B.White asked if the non-existence of an isoperimetric inequality for k-varifolds implies the existence of a nonzero, stationary, integral k-varifold. We present examples showing that this is not true in codimension greater than two. The …
A Simons' type formula for spacelike submanifolds in semi-Riemannian warped product
本文建立半黎曼扭曲乘积中类空子流形的 Simons 型公式，并据此推广平行与伪平行浸入结果，得出 de Sitter 与 Einstein-de Sitter 时空中的分类结论。
We prove long-time existence for mean curvature flow of a smooth $n$-dimensional spacelike submanifold of an $n+m$ dimensional manifold whose metric satisfies the timelike curvature condition.
Codimension-Two Spacelike Submanifolds with Umbilical Lightlike Normal Sections and Their Relationship to Lightlike Hypersurfaces
本文研究洛伦兹时空中具有脐性类光法向方向的余维二类空子流形，证明其受强几何拓扑约束，并建立分解定理：此类子流形可嵌入类光超曲面，且在全脐性条件下与共形平坦时空相关，为视界研究提供几何框架。
A world sheet in anti-de Sitter space is a timelike submanifold consisting of a one-parameter family of spacelike submanifolds. We consider the family of lightlike hypersurfaces along spacelike submanifolds in the world sheet. The locus of the singularities of lightlike hypersurfaces along spacelike submanifolds forms the caustic of the world sheet. This notion is originally introduced by Bousso and Randall in theoretical physics. In this paper we give a mathematical framework for the caustics of world sheets as an application of the theory of graph-like Legendrian unfoldings.
Lorentz polarisation and isoperimetric inequality in Minkowski spacetime
本文在Minkowski时空中引入Lorentz极化方法，证明了有限光锥依赖区域的等周不等式，揭示了类空超平面截断光锥在体积最大化中的最优性，并推广至双曲面和非时序超曲面的等周型不等式。
The classical Minkowski formula is extended to spacelike codimension-two submanifolds in spacetimes which admit "hidden symmetry" from conformal Killing-Yano two-forms. As an application, we obtain an Alexandrov type theorem for spacelike codimension-two submanifolds in a static spherically symmetric spacetime: a codimension-two submanifold with constant normalized null expansion (null mean curvature) must lie in a shear-free (umbilical) null hypersurface. These results are generalized for higher order curvature invariants. In particular, the notion of mixed higher order mean curvature is intr…
We prove spectral, stochastic and mean curvature estimates for complete $m$-submanifolds $φ\colon M \to N$ of $n$-manifolds with a pole $N$ in terms of the comparison isoperimetric ratio $I_{m}$ and the extrinsic radius $r_φ\leq \infty$. Our proof holds for the bounded case $r_φ< \infty$, recovering the known results, as well as for the unbounded case $r_φ=\infty$. In both cases, the fundamental ingredient in these estimates is the integrability over $(0, r_φ)$ of the inverse $I_{m}^{-1}$ of the comparison isoperimetric radius. When $r_φ=\infty$, this condition is guaranteed if $N$ is highly n…
The relative isoperimetric inequality for minimal submanifolds with free boundary in the Euclidean space
本文研究具自由边界的极小子流形的相对等周不等式，通过推广限制法锥思想得到最优面积估计，结合ABP方法给出新证明，并在余维数$m\leq2$时肯定回答了Choe提出的公开问题。
We obtain upper bounds for the isoperimetric quotients of extrinsic balls of submanifolds in ambient spaces which have a lower bound on their radial sectional curvatures. The submanifolds are themselves only assumed to have lower bounds on the radial part of the mean curvature vector field and on the radial part of the intrinsic unit normals at the boundaries of the extrinsic spheres, respectively. In the same vein we also establish lower bounds on the mean exit time for Brownian motion in the extrinsic balls. In those cases, where we may extend our analysis to hold all the way to infinity, we…
For Riemannian submanifolds of a semi-Riemannian manifold, we introduce the concepts of \emph{total shear tensor} and \emph{shear operators} as the trace-free part of the corresponding second fundamental form and shape operators. The relationship between these quantities and the umbilical properties of the submanifold is shown. Several novel notions of umbilical submanifolds are then considered along with the classical concepts of totally umbilical and pseudo-umbilical submanifolds. Then we focus on the case of co-dimension $2$, and we present necessary and sufficient conditions for the subman…
Given a semi-Riemannian manifold, we give necessary and sufficient conditions for a Riemannian submanifold of arbitrary co-dimension to be umbilical along normal directions. We do that by using the so-called \emph{total shear tensor}, i.e., the trace-free part of the second fundamental form. We define the \emph{shear space} and the \emph{umbilical space} as the spaces generated by the total shear tensor and by the umbilical vector fields, respectively. We show that the sum of their dimensions must equal the co-dimension.
New general results of non-existence and rigidity of spacelike submanifolds immersed in a spacetime, whose mean curvature is a time-oriented causal vector field, are given. These results hold for a wide class of spacetimes which includes globally hyperbolic, stationary, conformally stationary and pp-wave spacetimes, among others. Moreover, applications to the Cauchy problem in General Relativity, are presented. Finally, in the case of hypersurfaces, we also obtain significant consequences in Geometrical Analysis, solving new Calabi-Bernstein and Dirichlet problems on a Riemannian manifold.
Spacelike immersions in certain Lorentzian manifolds with lightlike foliations
本文研究广义 Schwarzschild 时空中位于类光超曲面叶内的类空子流形，给出平均曲率向量场公式并刻画其内外几何关系。
We consider the complex hyperbolic quadric ${Q^*}^n$ as a complex hypersurface of complex anti-de Sitter space. Shape operators of this submanifold give rise to a family of local almost product structures on ${Q^*}^n$, which are then used to define local angle functions on any Lagrangian submanifold of ${Q^*}^n$. We prove that a Lagrangian immersion into ${Q^*}^n$ can be seen as the Gauss map of a spacelike hypersurface of (real) anti-de Sitter space and relate the angle functions to the principal curvatures of this hypersurface. We also give a formula relating the mean curvature of the Lagran…
On the Geometry of Complete Spacelike LW-Submanifolds in Locally Symmetric Semi-Riemannian Spaces
研究完全类空LW子流形在局部对称半黎曼空间中的几何特性，通过刚性结果和分析技术推导出关于曲率的尖锐不等式，并统一了已知的分类定理。
Causal Invariance of the Contact Number of Pseudo-Riemannian Submanifolds
该研究证明伪黎曼子流形的普通、类空、类时接触数等价且重合，构造了接触数均为6的洛伦兹曲面，将无穷接触极限纳入半黎曼理论。
This is a survey of our work on spacelike graphic submanifolds in pseudo-Riemannian products, namely on Heinz-Chern and Bernstein-Calabi results and on the mean curvature flow, with applications to the homotopy of maps between Riemannian manifolds.
The optimal isoperimetric inequality for varifolds
本文证明了Brendle等周不等式对有限质量与全变差的varifolds成立，允许奇异部分，仅需可积平均曲率，常数在余维数一、二时最优。
Extremal and weakly trapped submanifolds in Generalized Robertson-Walker spacetimes
该研究针对广义Robertson-Walker时空中任意余维的类空子流形，借助最大值原理在抛物性假设下推导其刚性性质，进而实现对该时空下极值与弱俘获子流形的刻画。
We prove an inequality for submanifolds of Cartan-Hadamard manifolds, which relates the geometry of a submanifold to the measure of the geodesics in the ambient space which it intersects. For hypersurfaces, this gives an extension of Banchoff and Pohl's isoperimetric inequality to spaces of non-positive curvature. We also prove a modified version of Croke's isoperimetric inequality for hypersurfaces immersed in Cartan-Hadamard manifolds and a sharp, quantitative version of an isoperimetric inequality of Yau in spaces of negative curvature. We discuss the relationship between these results, and…
Submanifolds in Lorentz-Minkowski space are investigated from various mathematical viewpoints and are of interest also in relativity theory. We define the hyperbolic surface and the de Sitter surface of a curve in the spacelike hypersurface M in the Minkowski 4-space. These surfaces are respectively located in the hyperbolic 3-space and in the de Sitter 3-space. We use techniques of the theory of singularities in order to describe the generic shape of these surfaces and of their singular value sets. We also investigate geometric meanings of those singularities. The results in this paper contri…
A world sheet in Lorentz-Minkowski space is a timelike submanifold consisting of a one-parameter family of spacelike submanifolds in Lorentz-Minkowski space. In this paper we investigate differential geometry of world sheets in Lorentz-Minkowski space as an application of the theory of big wave fronts.
We prove explicit upper and lower bounds for the torsional rigidity of extrinsic domains of submanifolds P^m with controlled radial mean curvature in ambient Riemannian manifolds N^n with a pole p and with sectional curvatures bounded from above and from below, respectively. These bounds are given in terms of the torsional rigidities of corresponding Schwarz symmetrizations of the domains in warped product model spaces. Our main results are obtained using methods from previously established isoperimetric inequalities, as found in e.g. [MP4] and [MP5]. As in [MP4] we also characterize the geome…
In this work, we study the pseudo-Riemannian submanifolds of a pseudo-sphere with 1-type pseudo-spherical Gauss map. First, we classify the Lorentzian surfaces in a 4-dimensional pseudo-sphere $\mathbb{S}^4_s(1)$ with index s, $s=1, 2$, and having harmonic pseudo-spherical Gauss map. Then we give a characterization theorem for pseudo-Riemannian submanifolds of a pseudo-sphere $\mathbb{S}^{m-1}_s(1)\subset\mathbb{E}^m_s$ with 1-type pseudo-spherical Gauss map, and we classify spacelike surfaces and Lorentzian surfaces in the de Sitter space $\mathbb{S}^4_1(1)\subset\mathbb{E}^5_1$ with 1-type p…
On complete maximal submanifolds in pseudo-hyperbolic space
本文完整分类了伪双曲空间中的完备极大类空子流形，并探讨了其在Teichmüller理论与双曲群Anosov表示理论中的应用。
We prove that the length of the boundary of a $J$-holomorphic curve with Lagrangian boundary conditions is dominated by a constant times its area. The constant depends on the symplectic form, the almost complex structure, the Lagrangian boundary conditions and the genus. A similar result holds for the length of the real part of a real $J$-holomorphic curve. The infimum over $J$ of the constant properly normalized gives an invariant of Lagrangian submanifolds. We calculate this invariant to be $2π$ for the Lagrangian submanifold $\mathbb R P^n \subset \mathbb C P^n.$ We apply our result to prov…
Correspondence between Mean Curvature Flow and Harmonic-Ricci Flow
本文发现伪欧氏空间子流形的类空平均曲率流等价于对应高斯映射和诱导度量下耦合常数α=-1或+1的调和-里奇流，二者孤子也等价，并由此得到类空平均曲率流的单调性公式。
We give local, explicit representation formulas for n-dimensional spacelike submanifolds which are marginally trapped in the Minkowski space, the de Sitter and anti de Sitter spaces and the Lorentzian products of the sphere and the hyperbolic space by the real line.
Geometric inequalities and the Alexandrov-Bakelman-Pucci technique
本文基于Alexandrov-Bakelman-Pucci技术，统一讨论了几何不等式的证明框架，涵盖欧几里得空间的等周不等式、子流形的均曲率不等式及Sobolev不等式等核心方法与贡献。
The lightlike geometry of codimension two spacelike submanifolds in Lorentz-Minkowski space has been developed in [Izumiya, S. and Romero Fuster, M. C. Selecta Mathematica (NS), 13 23--55 (2007)] which is a natural Lorentzian analogue of the classical Euclidean differential geometry of hypersurfaces. In this paper we investigate a special class of surfaces (i.e., marginally trapped surfaces) in Minkowski space-time from the view point of the lightlike geometry.
Pólya--Szegö Inequality on Submanifolds of Riemannian Manifolds with Nonnegative Curvature and Applications
本文证明了非负曲率黎曼流形子流形上的Pólya--Szegö不等式，并由此推导出多种函数不等式，其常数与欧氏情形一致。
There are three types of hypersurfaces in a pseudoconformal space C^n_1 of Lorentzian signature: spacelike, timelike, and lightlike. These three types of hypersurfaces are considered in parallel. Spacelike hypersurfaces are endowed with a proper conformal structure, and timelike hypersurfaces are endowed with a conformal structure of Lorentzian type. Geometry of these two types of hypersurfaces can be studied in a manner that is similar to that for hypersurfaces of a proper conformal space. Lightlike hypersurfaces are endowed with a degenerate conformal structure. This is the reason that their…
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