Rigidity of invariant convex sets in symmetric spaces

Rigidity of compact Fuchsian manifolds with convex boundary

A compact Fuchsian manifold with boundary is a hyperbolic 3-manifold homeomorphic to $S_g \times [0; 1]$ such that the boundary component $S_g \times \{ 0\}$ is geodesic. We prove that a compact Fuchsian manifold with convex boundary is uniquely determined by the induced path metric on $S_g \times \{1\}$. We do not put further restrictions on the boundary except convexity.

凸多面体的标量曲率刚性

Scalar curvature rigidity of convex polytopes

本文证明了凸多面体的标量曲率刚性定理，方法上结合带边流形上Dirac算子的Fredholm理论与Fefferman-Phong型定理。

理想对称集的刚性

Rigidity of ideal symmetric sets

本文利用无穷远处对径集，在哈达玛空间中刻画了高秩模型几何（黎曼对称空间、欧几里得建筑和乘积空间）的刚性特征。

常曲率空间中凸超曲面的局部刚性

Local rigidity of convex hypersurfaces in spaces of constant curvature

本文证明了常曲率空间（$n\ge4$）中凸等距超曲面在严格凸点附近局部全等，推广了Senkin在$C^1$光滑假设下的结果。

Rigidity of mean convex subsets in non-negatively curved RCD spaces and stability of mean curvature bounds

We prove splitting theorems for mean convex open subsets in RCD (Riemannian curvature-dimension) spaces that extend results by Kasue, Croke and Kleiner for Riemannian manifolds with boundary to a non-smooth setting. A corollary is for instance Frankel's theorem. Then, we prove that our notion of mean curvature bounded from below for the boundary of an open subset is stable w.r.t. to uniform convergence of the corresponding boundary distance function. We apply this to prove almost rigidity theorems for uniform domains whose boundary has a lower mean curvature bound.

Entropy rigidity for finite volume strictly convex projective manifolds

We prove entropy rigidity for finite volume strictly convex projective manifolds in dimensions $\geq 3$, generalizing the work of arXiv:1708.03983 to the finite volume setting. The rigidity theorem uses the techniques of Besson, Courtois, and Gallot's entropy rigidity theorem. It implies uniform lower bounds on the volume of any finite volume strictly convex projective manifold in dimensions $\geq 3$.

Hopf rigidity for convex billiards on the hemisphere and hyperbolic plane

This paper deals with Hopf type rigidity for convex billiards on surfaces of constant curvature. We prove that the only convex billiard without conjugate points on the Hyperbolic plane or on the Hemisphere is circular billiard.

On the infinitesimal rigidity of polyhedra with vertices in convex position

Let $P \subset \R^3$ be a polyhedron. It was conjectured that if $P$ is weakly convex (i. e. its vertices lie on the boundary of a strictly convex domain) and decomposable (i. e. $P$ can be triangulated without adding new vertices), then it is infinitesimally rigid. We prove this conjecture under a weak additional assumption of codecomposability. The proof relies on a result of independent interest concerning the Hilbert-Einstein function of a triangulated convex polyhedron. We determine the signature of the Hessian of that function with respect to deformations of the interior edges. In partic…

凸实射影流形的一个高阶刚性定理

A higher rank rigidity theorem for convex real projective manifolds

本文研究凸实射影流形的刚性问题，将 Ballmann 与 Burns-Spatzier 的高阶刚性定理推广到该情形，证明了相应类似定理。

On the local rigidity of Einstein manifolds with convex boundary

Let (M, g) be a compact Einstein manifold with non-empty boundary. We prove that Killing fields at the boundary extend to Killing fields of any (M, g) provided the boundary is weakly convex and a simple condition on the fundamental group holds. This gives a new proof of the classical infinitesimal rigidity of convex surfaces in Euclidean space and generalizes the result to Einstein metrics of any dimension.

Rigidity of Entire Convex Self-Shrinking Solutions to Hessian Quotient Flows

We prove that all entire smooth strictly convex self-shrinking solutions on $\mathbb{R}^n$ to the Hessian quotient flows must be quadratic. This generalizes the rigidity theorem for entire self-shrinking solutions to the Lagrangian mean curvature flow in pseudo-Euclidean space due to Ding-Xin \cite{DX}. Moreover, we show that our argument works for a larger class of equations. In particular, we obtain rigidity results for entire self-shrinking solutions on $\mathbb{C}^n$ to the Kähler-Ricci flow under certain conditions.

具有凸边界的反德西特(2+1)时空在Fuchsian轨迹附近的刚性

Rigidity of anti-de Sitter (2+1)-spacetimes with convex boundary near the Fuchsian locus

本文证明了在Fuchsian轨迹附近，具有严格凸类空边界（光滑或多面体）的整体双曲紧致反德西特(2+1)时空由边界诱导度量唯一确定。

On the infinitesimal rigidity of weakly convex polyhedra

The main motivation here is a question: whether any polyhedron which can be subdivided into convex pieces without adding a vertex, and which has the same vertices as a convex polyhedron, is infinitesimally rigid. We prove that it is indeed the case for two classes of polyhedra: those obtained from a convex polyhedron by ``denting'' at most two edges at a common vertex, and suspensions with a natural subdivision.

A rigidity criterion for non-convex polyhedra

Let $P$ be a (non necessarily convex) embedded polyhedron in $\R^3$, with its vertices on an ellipsoid. Suppose that the interior of $P$ can be decomposed into convex polytopes without adding any vertex. Then $P$ is infinitesimally rigid. More generally, let $P$ be a polyhedron bounding a domain which is the union of polytopes $C_1, ..., C_n$ with disjoint interiors, whose vertices are the vertices of $P$. Suppose that there exists an ellipsoid which contains no vertex of $P$ but intersects all the edges of the $C_i$. Then $P$ is infinitesimally rigid. The proof is based on some geometric prop…

双曲空间中抛物凸多面体的数量曲率刚性

Scalar curvature rigidity of parabolic convex polytopes in hyperbolic space

本文利用旋量技术与Brendle-Wang平滑构造，证明了双曲空间中抛物凸多面体的数量曲率刚性，并进一步得出有界光滑区域的Llarull型刚性与多面体初始数据集的二面角刚性。

欧几里得空间中严格凸超曲面与具有非零欧拉特征的非负曲率空间乘积的标量曲率刚性

Scalar Curvature Rigidity for Products of convex Hypersurfaces and even dimensional Manifolds

研究欧几里得空间中严格凸超曲面与特定非负曲率空间乘积的标量曲率刚性，基于弗雷德霍姆族指标定理给出证明，恢复了相关结果。

Infinitesimal rigidity of convex surfaces through the second derivative of the Hilbert-Einstein functional II: Smooth case

The paper is centered around a new proof of the infinitesimal rigidity of smooth closed surfaces with everywhere positive Gauss curvature. We use a reformulation that replaces deformation of an embedding by deformation of the metric inside the body bounded by the surface. The proof is obtained by studying derivatives of the Hilbert-Einstein functional with boundary term. This approach is in a sense dual to proving the Gauss infinitesimal rigidity, that is rigidity with respect to the Gauss curvature parametrized by the Gauss map, by studying derivatives of the volume bounded by the surface. We…

On entropy, regularity and rigidity for convex representations of hyperbolic manifolds

Given a convex representation $ρ:Γ\to\textrm{PGL}(d,\mathbb{R})$ of a convex co-compact group $Γ$ of $\mathbb{H}^k$ we find upper bounds for the quantity $αh_ρ,$ where $h_ρ$ is the entropy of $ρ$ and $α$ is the Hölder exponent of the equivariant map $\partialΓ\to\mathbb{P}(\mathbb{R}^d).$ We also give rigidity statements when the upper bound is attained. We then study Hitchin representations and prove that if $ρ:π_1Σ\to\textrm{PSL}(d,\mathbb{R})$ is in the Hitchin component then $αh_ρ\leq 2/(d-1)$ (where $α$ is the Hölder exponent of the map $ζ:\partial\mathbb{H}^2\to\mathscr{F}$) with equalit…

Rigidity of convex divisible domains in flag manifolds

In contrast to the many examples of convex divisible domains in real projective space, we prove that up to projective isomorphism there is only one convex divisible domain in the Grassmannian of $p$-planes in $\mathbb{R}^{2p}$ when $p > 1$. Moreover, this convex divisible domain is a model of the symmetric space associated to the simple Lie group SO$(p, p)$.

Cartan-Hadamard流形中超曲面的凸性与刚性

Convexity and rigidity of hypersurfaces in Cartan-Hadamard manifolds

本文证明Cartan-Hadamard流形中曲率在切平面消失的闭无穷小凸超曲面围成凸平坦区域，推广了经典刻画与刚性结果，并应用Gauss-Codazzi方程和Alexandrov几何技巧。

Rigidity of entire self-shrinking solutions to Kähler-Ricci flow with strictly real convex potential

We prove that every entire self-shrinking solution on $\mathbb{C}^n$ to the Kähler-Ricci flow with strictly real convex potential must be quadratic. The very same argument also gives a pointwise proof for the rigidity of entire self-shrinking solutions to Lagrangian mean curvature flow in pseudo-Euclidean space obtained by Q. Ding and Y.L. Xin. Furthermore, we show that our argument works for a larger class of equations.

Rigidity of area-minimizing free boundary surfaces in mean convex three-manifolds

We prove a local splitting theorem for three-manifolds with mean convex boundary and scalar curvature bounded from below that contain certain locally area-minimizing free boundary surfaces. Our methods are based on those of Micallef and Moraru. We use this local result to establish a global rigidity theorem for area-minimizing free boundary disks. In the negative scalar curvature case, this global result implies a rigidity theorem for solutions of the Plateau problem with length-minimizing boundary.

凸 NNSC 流形中完备自由边界极小超曲面的刚性

Rigidity of Complete Free Boundary Minimal Hypersurfaces in Convex NNSC Manifolds

本文证明了在 R^4 单位球中不存在完备的双侧稳定自由边界浸入，该结论源于利用翘曲 θ-气泡方法得出的在凸边界和非负中间 Ricci 曲率完备 4 维流形中极小超曲面的刚性定理。

凸体在积分流空间中的填充极小性与Lipschitz-体积刚性

Filling minimality and Lipschitz-volume rigidity of convex bodies among integral current spaces

本文证明了凸体在积分流空间中是其边界度量的唯一极小填充，并具有Lipschitz-体积刚性性质，同时回答了关于Plateau问题极小化序列内蕴平坦收敛的Perales问题。

Boundary and lens rigidity for non-convex manifolds

We study the boundary and lens rigidity problems on domains without assuming the convexity of the boundary. We show that such rigidities hold when the domain is a simply connected compact Riemannian surface without conjugate points. For the more general class of non-trapping compact Riemannian surfaces with no conjugate points, we show lens rigidity. We also prove the injectivity of the X-ray transform on tensors in a variety of settings with non-convex boundary and, in some situations, allowing a non-empty trapped set.

Rigidity of complex convex divisible sets

An open convex set in real projective space is called divisible if there exists a discrete group of projective automorphisms which acts co-compactly. There are many examples of such sets and a theorem of Benoist implies that many of these examples are strictly convex, have $C^1$ boundary, and have word hyperbolic dividing group. In this paper we study a notion of convexity in complex projective space and show that the only divisible complex convex sets with $C^1$ boundary are the projective balls.

Rigidity of Busemann convex Finsler metrics

We prove that a Finsler metric is nonpositively curved in the sense of Busemann if and only if it is affinely equivalent to a Riemannian metric of nonpositive sectional curvature. In other terms, such Finsler metrics are precisely Berwald metrics of nonpositive flag curvature. In particular in dimension 2 every such metric is Riemannian or locally isometric to that of a normed plane. In the course of the proof we obtain new characterizations of Berwald metrics in terms of the so-called linear parallel transport.

Projective Anosov representations, convex cocompact actions, and rigidity

In this paper we show that many projective Anosov representations act convex cocompactly on some properly convex domain in real projective space. In particular, if a non-elementary word hyperbolic group is not commensurable to a non-trivial free product or the fundamental group of a closed hyperbolic surface, then any projective Anosov representation of that group acts convex cocompactly on some properly convex domain in real projective space. We also show that if a projective Anosov representation preserves a properly convex domain, then it acts convex cocompactly on some (possibly different)…

平均凸三维流形中自由边界极小圆盘的刚性

Rigidity of free boundary minimal disks in mean convex three-manifolds

本文研究平均凸、标量曲率有正下界的黎曼三维流形中局部最大化修正 Hawking 质量的自由边界极小圆盘，并在严格稳定条件下证明其邻域具有半 de Sitter–Schwarzschild 刚性。

关于紧区域标量曲率比较刚性的一个注记

A Note on Scalar curvature comparison rigidity for compact domains

本文借助 Brendle 边界条件下的调和旋量，将 Gromov 凸多面体标量曲率刚性猜想推广至任意凸 Riemann 多面体型区域，并得到与 Shi-Tam 结果平行的曲率比较刚性定理。

Convex real projective structures and Weil's local rigidity Theorem

For an $n$-dimensional real hyperbolic manifold $M$, we calculate the Zariski tangent space of a character variety $χ(π_1(M),SL(n+1,\mathbb R)), n>2$ at Fuchisan loci to show that the tangent space consists of cubic forms. Furthermore we prove the Weil's local rigidity theorem for uniforml hyperbolic lattices using real projective structures.

Volume entropy rigidity of non-positively curved symmetric spaces

We characterize symmetric spaces of non-positive curvature by the equality case of general inequalities between geometric quantities

On the Rigidity of Spherically Symmetric Finsler Metrics with Isotropic $E$-Curvature

In the current paper, first we give the correct version of the formula for mean Berwald curvature of a spherically symmetric Finsler metric given in paper \cite{YCheWSon2015}. Further, we establish differential equations characterizing projectively as well as dually flat spherically symmetric Finsler metrics. Finally, we obtain a rigidity result on spherically symmetric Finsler metrics with isotropic $E$-Curvature.

Rigidity of Rank-One Factors of Compact Symmetric Spaces

We consider the decomposition of a compact-type symmetric space into a product of factors and show that the rank-one factors, when considered as totally geodesic submanifolds of the space, are isolated from inequivalent minimal submanifolds.

Rank one symmetric spaces and Rigidity

In this paper we show that if the limit set is not small ,marked length spectrum determines geometric structure of rank one locally symmetric manifolds.

关于紧致赫尔米特对称空间中间隙刚性问题

On gap rigidity problems for compact Hermitian symmetric spaces

本文研究了紧致赫尔米特对称空间中对角曲线的间隙刚性问题，并给出了更高维子流形的更弱间隙刚性定理。

Filling volume minimality and boundary rigidity of metrics close to a negatively curved symmetric metric

This paper generalizes D. Burago and S. Ivanov's work on filling volume minimality and boundary rigidity of almost real hyperbolic metrics. We show that regions with metrics close to a negatively curved symmetric metric are strict minimal fillings and hence boundary rigid. This includes perturbations of complex, quaternionic and Cayley hyperbolic metrics.

标量曲率比较与3维弱凸区域的刚性

Scalar curvature comparison and rigidity of $3$-dimensional weakly convex domains

本文证明了具有平均凸边界的紧致黎曼3-流形在标量曲率非负且满足缩放平均曲率比较时是平坦的，并将结果推广至带角流形，其证明利用了毛细极小曲面和叶状结构。

Minimal entropy rigidity for lattices in products of rank one symmetric spaces

We prove minimal entropy rigidity for complete, finite volume manifolds locally isometric to a product of rank one symmetric spaces of dimension at least 3: the locally symmetric metric uniquely minimizes (normalized) entropy among all Riemannian metrics. The corresponding theorem is true for maps into these spaces as well.

Lens Rigidity in Scattering by Unions of Strictly Convex Bodies in $\R^2$

It was proved in \cite{NS1} that obstacles $K$ in $\R^d$ that are finite disjoint unions of strictly convex domains with $C^3$ boundaries are uniquely determined by the travelling times of billiard trajectories in their exteriors and also by their so called scattering length spectra. However the case $d = 2$ is not properly covered in \cite{NS1}. In the present paper we give a separate different proof of the same result in the case $d = 2$.

Rigidity phenomena on lower $N$-weighted Ricci curvature bounds with $\varepsilon$-range for non-symmetric Laplacian

Lu-Minguzzi-Ohta have introduced the notion of a lower $N$-weighted Ricci curvature bound with $\varepsilon$-range, and derived several comparison geometric estimates from a Laplacian comparison theorem for weighted Laplacian. The aim of this paper is to investigate various rigidity phenomena for the equality case of their comparison geometric results. We will obtain rigidity results concerning the Laplacian comparison theorem, diameter comparisons, and volume comparisons. We also generalize their works for non-symmetric Laplacian induced from vector field potential.

双曲三维空间中凸曲面的Weyl问题的刚性定理

Rigidity Theorems for the Weyl Problem of Convex Surfaces in Hyperbolic 3-Space

研究双曲三维空间中非紧凸集的刚性，证明若两个非紧闭凸子集的理想边界为圆型闭集且连通分支可数，则边界间的内蕴等距可延拓为整体等距，推广了Luo-Luo-Rao的刚性定理，并建立了Weyl问题的唯一性结果。

On the rigidity of mean convex self-shrinkers

Self-shrinkers model singularities of the mean curvature flow; they are defined as the special solutions that contract homothetically under the flow. Colding-Ilmanen-Minicozzi showed that cylindrical self-shrinkers are rigid in a strong sense - that is, any self-shrinker that is mean convex with uniformly bounded curvature on a large, but compact, set must be a round cylinder. Using this result, Colding and Minicozzi were able to establish uniqueness of blowups at cylindrical singularities, and provide a detailed description of the singular set of generic mean curvature flows. In this paper, w…

Griffiths-Harris rigidity of compact Hermitian symmetric spaces

I show that any complex manifold that resembles a rank two compact Hermitian symmetric space (other than a quadric hypersurface) to order two at a general point must be an open subset of such a space.

Entropy rigidity of symmetric spaces without focal points

We characterize symmetric spaces without focal points by the equality case of general equalities between geometric quantities.

Lens rigidity with partial data in the presence of a magnetic field

In this paper we consider the lens rigidity problem with partial data for conformal metrics in the presence of a magnetic field on a compact manifold of dimension $\geq 3$ with boundary. We show that one can uniquely determine the conformal factor and the magnetic field near a strictly convex (with respect to the magnetic geodesics) boundary point where the lens data is accessible. We also prove a boundary rigidity result with partial data assuming the lengths of magnetic geodesics joining boundary points near a strictly convex boundary point are known. The local lens rigidity result also lead…

Rigidity and quasi-rigidity of extremal cycles in Hermitian symmetric spaces

I use local differential geometric techniques to prove that the algebraic cycles in certain extremal homology classes in Hermitian symmetric spaces are either rigid (i.e., deformable only by ambient motions) or quasi-rigid (roughly speaking, foliated by rigid subvarieties in a nontrivial way). These rigidity results have a number of applications: First, they prove that many subvarieties in Grassmannians and other Hermitian symmetric spaces cannot be smoothed (i.e., are not homologous to a smooth subvariety). Second, they provide characterizations of holomorphic bundles over compact Kahler mani…

平坦与球面背景下具有平均凸边界的流形的 Brown-York 质量正定性及刚性的紧致方法

Compact approach to the positivity of Brown-York mass and rigidity of manifolds with mean-convex boundaries in flat and spherical contexts

本文利用旋量证明与 ADM 质量正定性，为 Brown-York 质量正定性提供了紧致方法，并得出 Dirac 算子特征值下界及平坦与球面空间中平均凸流形的刚性结果。

具有局部对称叶的李叶状结构的刚性

Rigidity of Lie foliations with locally symmetric leaves

本文证明极小李叶状结构的叶若局部等距于无庞加莱圆盘因子的非紧型对称空间，则光滑共轭于齐次李叶状结构，推广了Zimmer定理，并应用于和乐群算术性与黎曼叶状结构刚性。

Cartan-Hadamard流形中曲面的总绝对曲率与刚性

Total absolute curvature and rigidity of surfaces in Cartan-Hadamard manifolds

研究Cartan-Hadamard流形中总绝对曲率最小的闭曲面，证明其边界为平坦凸体，并推广Chern-Lashof定理，解决Gromov于1985年提出的问题。

Boundary rigidity with partial data

We study the boundary rigidity problem with partial data consisting of determining locally the Riemannian metric of a Riemannian manifold with boundary from the distance function measured at pairs of points near a fixed point on the boundary. We show that one can recover uniquely and in a stable way a conformal factor near a strictly convex point where we have the information. In particular, this implies that we can determine locally the isotropic sound speed of a medium by measuring the travel times of waves joining points close to a convex point on the boundary. The local results lead to a g…

Margulis 多重宇宙：无穷小刚性、压力形式与凸性

Margulis Multiverse: Infinitesimal Rigidity, Pressure Form and Convexity

本文研究高维无尖点 Margulis 时空模空间，构造压力形式，借助带标记 Margulis 不变量谱的无穷小确定性证明其在恒定熵截面上诱导黎曼度量，并确立固定线性部分时这些截面的凸性。

Deformations of hyperbolic convex polyhedra and 3-cone-manifolds

The Stoker problem, first formulated in 1968, consists in understanding to what extent a convex polyhedron is determined by its dihedral angles. By means of the double construction, this problem is intimately related to rigidity issues for 3-dimensional cone-manifolds. In a former paper, two such rigidity results were proven, implying that the infinitesimal version of the Stoker conjecture is true in the hyperbolic and Euclidean cases. In this second article, we prove that local rigidity holds and obtain that the space of convex hyperbolic polyhedra with given combinatorial type is locally par…

无限圆图案和双曲三维空间中凸多面体的特征化

Characterizations of infinite circle patterns and convex polyhedra in hyperbolic 3-space

本文研究了双曲三维空间中无限圆图案和凸多面体的特征化问题，证明了当0≤Θ<π时无限圆图案的刚性和统一化定理，并解决了正则圆图案的分类问题。