Masoud Sabzevari, Amir Hashemi
In this paper, we prove that every real analytic totally nondegenerate model CR manifold of length >= 3 has rigidity. This result was actually conjectured before by Valerii Beloshapka as the so-called "maximum conjecture". It follows that the transformation Lie group of all CR automorphisms associated with each of the mentioned models does not include any nonlinear map.
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It is shown that two Levi-Tanaka and infinitesimal CR automorphism algebras, associated with a totally nondegenerate model of CR dimension one are isomorphic. As a result, the model surfaces are maximally homogeneous and standard. This gives an affirmative answer in CR dimension one to a certain question formulated by Beloshapka.
Applying the equivariant moving frames method, we construct convergent normal forms for real-analytic 5-dimensional totally nondegenerate CR submanifolds of C^4. These CR manifolds are divided into several biholomorphically inequivalent subclasses, each of which has its own complete normal form. Moreover it is shown that, biholomorphically, Beloshapka's cubic model is the unique member of this class with the maximum possible dimension seven of the corresponding algebra of infinitesimal CR automorphisms. Our results are also useful in the study of biholomorphic equivalence problem between CR ma…
Applying Elie Cartan's classical method, we show that the biholomorphic equivalence problem to a totally nondegenerate Beloshapka's model of CR dimension one and codimension $k> 1$, whence of real dimension $2+k$, is reducible to some absolute parallelism, namely to an {e}-structure on a certain prolonged manifold of real dimension either $3+k$ or $4+k$. The proof relies on the weight analysis of the structure equations associated with the mentioned problem of equivalence. Thanks to the achieved results, we prove Beloshapka's maximum conjecture about the rigidity of his CR models of certain le…
A CR manifold $M$, with CR distribution $\mathcal D^{10}\subset T^\mathbb C M$, is called {\it totally nondegenerate of depth $μ$} if: (a) the complex tangent space $T^\mathbb C M$ is generated by all complex vector fields that might be determined by iterated Lie brackets between at most $μ$ fields in $\mathcal D^{10} + \overline{\mathcal D^{10}}$; (b) for each integer $2 \leq k \leq μ-1$, the families of all vector fields that might be determined by iterated Lie brackets between at most $k$ fields in $\mathcal D^{10} + \overline{\mathcal D^{10}}$ generate regular complex distributions; (c) th…
A well known Conjecture due to Beloshapka asserts that all totally nondegenerate polynomial models with the length $l\geq 3$ of their Levi-Tanaka algebra are {\em rigid}, that is, any point preserving automorphism of them is completely determined by the restriction of its differential at the fixed point onto the complex tangent space. For the length $l=3$, Beloshapka's Conjecture was proved by Gammel and Kossovskiy in 2006. In this paper, we prove the Conjecture for arbitrary length $l\geq 3$. As another application of our method, we construct polynomial models of length $l\geq 3$, which are n…
Rigidity and vanishing results on totally real submanifolds under $L^p$-integrable conditions
本文修正并推广了Cuong等人在复空间形式中全实极小子流形上的刚性与消失性结果，主要贡献是将$L^p$可积条件中$p$的取值范围进行了扩展。
Constructions and Rigidity of Nondegenerate $\mathbb{Z}_2$ Harmonic Functions with Quadric Branching Sets
本文研究欧氏空间上分支轨迹为椭球面、平面二次曲线的非退化$\boldsymbol{\rm Z}_2$调和函数，证明相关参数映射性质、构造特定分支轨迹的调和函数并揭示其与特殊拉格朗日三维流形的关联。
Let $Q^N_l\subset \bC\bP^{N+1}$ denote the standard real, nondegenerate hyperquadric of signature $l$ and $M\subset \bC^{n+1}$ a real, Levi nondegenerate hypersurface of the same signature $l$. We shall assume that there is a holomorphic mapping $H_0\colon U\to \bC\bP^{N_0+1}$, where $U$ is some neighborhood of $M$ in $\bC^{n+1}$, such that $H_0(M)\subset Q^{N_0}_l$ but $H(U)\not\subset Q^{N_0}_l$. We show that if $N_0-n<l$ then, for any $N\geq N_0$, any holomorphic mapping $H\colon U\to \bC\bP^{N+1}$ with $H(M)\subset Q^{N}_l$ and $H(U)\not\subset Q^{N_0}_l$ must be the standard linear embedd…
We consider a totally nonsymplectic Anosov action of Z^k which is either uniformly quasiconformal or pinched on each coarse Lyapunov distribution. We show that such an action on a torus is C^\infty--conjugate to an action by affine automorphisms. We also obtain similar global rigidity results for actions on an arbitrary compact manifold assuming that the coarse Lyapunov foliations are jointly integrable.
Formation and structural stability of nondegenerate neckpinches
本文证明平均曲率流中非退化颈缩奇点在有限时间内形成，给出其精确渐近轮廓，并证明其在C^2扰动下结构稳定，结合密度定理表明该现象在三维中普遍且稳定。
We obtain a bound for the area of a capillary $H-$surface in a three-manifold with umbilic boundary and controlled sectional curvature. We then analyze the geometry when this area bound is realized, and obtain rigidity theorems. As a side product, we obtain existence of totally geodesic embedded surfaces in hyperbolic three-manifolds under the assumption of the existence of a $H-$surface realizing the area bound, in particular, under the existence of a totally umbilic $H-$surface.
Some rigidity results on compact hypersurfaces with capillary boundary in Hyperbolic space
本文证明了双曲空间中毛细超曲面的Heintze-Karcher不等式，仅在完全脐曲超曲面时成立，并应用该结果证明了嵌入毛细超曲面的Alexandrov型定理，还证明了在双曲空间中支撑于全测地平面的毛细超曲面的其他刚性结果。
Motivated by the ideas and methods used by Naitoh in the consideration of parallel totally real submanifolds in complex space forms, the author of the present paper successfully makes use of the so called Jordan triple and (restricted) structure Lie algebra associated with a given Jordan algebra to establish a one-to-one correspondence between the set of equivalence classes of connected, simply connected and nondegenerate equiaffine symmetric hypersurfaces with a given nonzero affine mean curvature and that of the equivalence classes of semi-simple real Jordan algebras. Then, via the existing …
Let $N\subset GL(n,R)$ be the group of upper triangular matrices with $1$s on the diagonal, equipped with the standard Carnot group structure. We show that quasiconformal homeomorphisms between open subsets of $N$, and more generally Sobolev mappings with nondegenerate Pansu differential, are rigid when $n \geq 4$; this settles the Regularity Conjecture for such groups. This result is deduced from a rigidity theorem for the manifold of complete flags in $R^n$. Similar results also hold in the complex and quaternion cases.
Rigidity of Totally Geodesic Hypersurfaces in Negative Curvature
本文研究负曲率流形中全测地超曲面在体积最小化等号情形下是否迫使流形等距，证明许多超曲面具有刚性，并构造反例。
We prove the following rigidity theorem: For an n-dimensional compact Riemannian manifold with boundary whose Ricci curvature is bounded by n-1 from below, if its boundary is isometric to the standard sphere of dimension n-1 and totally geodesic, then the manifold is isometric to the standard hemisphere.
We prove each embedded, constant mean curvature (CMC) surface in Euclidean space with genus zero and finitely many coplanar ends is nondegenerate: there is no nontrivial square-integrable solution to the Jacobi equation, the linearization of the CMC condition. This implies that the moduli space of such coplanar surfaces is a real-analytic manifold and that a neighborhood of these in the full CMC moduli space is itself a manifold. Nondegeneracy further implies (infinitesimal and local) rigidity in the sense that the asymptotes map is an analytic immersion on these spaces, and also that the copl…
We prove a Milnor-Wood inequality for representations of the fundamental group of a compact complex hyperbolic manifold in the group of isometries of quaternionic hyperbolic space. Of special interest is the case of equality, and its application to rigidity. We show that equality can only be achieved for totally geodesic representations, thereby establishing a global rigidity theorem for totally geodesic representations.
Let W be a compact hyperbolic n-manifold with totally geodesic boundary. We prove that if n>3 then the holonomy representation of pi_1 (W) into the isometry group of hyperbolic n-space is infinitesimally rigid.
We study Riemannian foliations with complex leaves on Kaehler manifolds. The tensor T, the obstruction to the foliation be totally geodesic, is interpreted as a holomorphic section of a certain vector bundle. This enables us to give classification results when the manifold is compact.
Let $M$ be an $n(\geq3)$-dimensional oriented compact submanifold with parallel mean curvature in the simply connected space form $F^{n+p}(c)$ with $c+H^2>0$, where $H$ is the mean curvature of $M$. We prove that if the Ricci curvature of $M$ satisfies $Ric_{M}\geq(n-2)(c+H^2),$ then $M$ is either a totally umbilic sphere, the Clifford hypersurface $S^{m}\big(\frac{1}{\sqrt{2(c+H^2)}}\big)\times S^{m}\big(\frac{1}{\sqrt{2(c+H^2)}}\big)$ in $S^{n+1}(\frac{1}{\sqrt{c+H^2}})$ with $n=2m$, or $\mathbb{C}P^{2}(4/3(c+H^2))$ in $S^7(\frac{1}{\sqrt{c+H^2}})$. In particular, if $Ric_{M}>(n-2)(c+H^2),$ …
In this paper we establish conditions on the length of the second fundamental form of a complete minimal submanifold $M^n$ in the hyperbolic space $\mathbb{H}^{n+m}$ in order to show that $M^n$ is totally geodesic. We also obtain sharp upper bounds estimates for the first eigenvalue of the super stability operator in the case of $M$ is a surface in $\mathbb{H}^{4}$.
In this short note, we show the rigidity of a trace estimate for Steklov eigenvalues with respect to functions in our previous work (Trace and inverse trace of Steklov eigenvalues. J. Differential Equations 261 (2016), no. 3, 2026--2040.). Namely, we show that equality of the estimate holds if and only if the manifold is a direct product of a round ball and a closed manifold. The key ingredient in the proof is a decomposition theorem for flat and totally geodesic Riemannian submersions which may be of independent interests.
Second Eigenvalue Estimates and Spectral Rigidity for Submanifolds of Compact Rank-One Symmetric Spaces
该研究针对紧射影空间及凯莱射影平面上的闭子流形，建立薛定谔算子第二特征值的上界，推导普适估计，证明等式对应全脐子流形，确定达界的全测地例子。
Real analytic ($\mathcal{C}^ω$) surfaces $S^2$ in $\mathbb{R}^3 \ni (x,y,u)$ graphed as $\big\{ u = F(x,y) \big\}$ with $F_{xx} \neq 0$ whose Gaussian curvature vanishes identically: \[ 0 \,\equiv\, F_{xx}\,F_{yy} - F_{xy}^2, \] possess, under the action of the affine transformation group ${\sf Aff}_3(\mathbb{R}) = {\sf GL}_3(\mathbb{R}) \ltimes \mathbb{R}^3$, a basic invariant analogous to $2$-nondegeneracy for $\mathcal{C}^ω$ real hypersurfaces $M^5 \subset \mathbb{C}^3$: \[ S_{\sf aff} \,:=\, \frac{F_{xx}\,F_{xxy}-F_{xy}\,F_{xxx}}{ F_{xx}^2}. \] It is known (or easily recovered) that $S$ is…
In this paper we prove an area comparison result for certain totally geodesic surfaces in 3-manifolds with a lower bound on the scalar curvature. This result is a variant of a comparison theorem of Heintze-Karcher for minimal hypersurfaces in manifolds of nonnegative Ricci curvature. Our assumptions on the ambient manifold are weaker but the assumptions on the surface are considerably more restrictive. We then use our comparison theorem to provide a unified proof of various splitting theorems for 3-manifolds with lower bounds on the scalar curvature.
For n>3 we study spaces obtained from finite volume complete real hyperbolic n-manifolds by removing a compact totally geodesic submanifold of codimension two. We prove that their fundamental groups are relative hyperbolic, co-Hopf, biautomatic, residually hyperbolic, not Kähler, not isomorphic to lattices in virtually connected real Lie groups, have no nontrivial subgroups with property (T), have finite outer automorphism groups, satisfy Mostow-type Rigidity, have finite asymptotic dimension and rapid decay property, and satisfy Baum-Connes conjecture. We also characterize those lattices in r…
We prove global rigidity for compact hyperbolic and spherical cone-3-manifolds with cone-angles $\leq π$ (which are not Seifert fibered in the spherical case), furthermore for a class of hyperbolic cone-3-manifolds of finite volume with cone-angles $\leq π$, possibly with boundary consisting of totally geodesic hyperbolic turnovers. To that end we first generalize our local rigidity result to the setting of hyperbolic cone-3-manifolds of finite volume as above. We then use the geometric techniques developed by Boileau, Leeb and Porti to deform the cone-manifold structure to a complete non-sing…
A contact stationary Legendrian submanifold (briefly, CSL submanifold) is a stationary point of the volume functional of Legendrian submanifolds in a Sasakian manifold. Much effort has been paid in the last two decades to construct examples of such manifolds, mainly by geometers using various geometric methods. But we have rare knowledge about their geometric properties till now. Recently, Y. Luo (\cite{ Luo2, Luo1}) proved that a closed CSL surface in $\mathbb{S}^5$ with the square length of its second fundamental form belonging to $[0,2]$ must be totally geodesic or be a flat minimal Legendr…
In this paper, we prove a version of the classical Cartan-Hadamard theorem for negatively curved manifolds, of dimension $n\neq 5$, with non-empty totally geodesic boundary. More precisely, if $M_1^n,M_2^n$ are any two such manifolds, we show that (1) $\partial ^\infty \tilde M_1^n$ is homeomorphic to $\partial ^\infty \tilde M_2^n$, and (2) $\tilde M_1^n$ is homeomorphic to $\tilde M_2^n$. As a sample application, we show that simple, thick, negatively curved P-manifolds of dimension $\geq 6$ are topologically rigid. We include some straightforward consequences of topological rigidity (diagra…
Localized Curvature Domination and Rigidity of Harmonic Maps
该研究通过局部曲率支配原理，证明了调和映射在特定曲率条件下为常数，并给出了临界阈值下的相似分类。
On Umbilical Real Hypersufaces of Products of Complex Space Forms
研究复空间形式积中的脐实超曲面的刚性与不存在性，证明无局部积结构的超曲面形算子不平行，分类显示具有局部几乎积结构的超曲面必为全测地或内蕴超球面。
We study the local equivalence problem for real-analytic ($\mathcal{C}^ω$) hypersurfaces $M^5 \subset \mathbb{C}^3$ which, in coordinates $(z_1, z_2, w) \in \mathbb{C}^3$ with $w = u+i\, v$, are rigid: \[ u \,=\, F\big(z_1,z_2,\overline{z}_1,\overline{z}_2\big), \] with $F$ independent of $v$. Specifically, we study the group ${\sf Hol}_{\sf rigid}(M)$ of rigid local biholomorphic transformations of the form: \[ \big(z_1,z_2,w\big) \longmapsto \Big( f_1(z_1,z_2), f_2(z_1,z_2), a\,w + g(z_1,z_2) \Big), \] where $a \in \mathbb{R} \backslash \{0\}$ and $\frac{D(f_1,f_2)}{D(z_1,z_2)} \neq 0$, whic…
We consider isometric immersions of complete connected Riemannian manifolds into space forms of nonzero constant curvature. We prove that if such an immersion is compact and has semi-definite second fundamental form, then it is an embedding with codimension one, its image bounds a convex set, and it is rigid. This result generalizes previous ones by M. do Carmo and E. Lima, as well as by M. do Carmo and F. Warner. It also settles affirmatively a conjecture by do Carmo and Warner. We establish a similar result for complete isometric immersions satisfying a stronger condition on the second funda…
We show that the focal radius of any submanifold $N$ of positive dimension in a manifold $M$ with sectional curvature greater than or equal to $1$ does not exceed $\frac{π}{2}.$ In the case of equality, we show that $N$ is totally geodesic in $M$ and the universal cover of $M$ is isometric to a sphere or a projective space with their standard metrics, provided $N$ is closed. Our results also hold for $k^{th}$--intermediate Ricci curvature, provided the submanifold has dimension $\geq k.$ Thus in a manifold with Ricci curvature $\geq n-1,$ all hypersurfaces have focal radius $\leq \frac{π}{2},$…
On the invariant trajectories of lightlike paths near central potential singularities in semi-Riemannian manifolds of index 2
本文探讨了三维半黎曼流形中完全脐子流形上null螺旋的几何性质，通过分析null弗莱内框架和退化度量条件，推导出表征null螺旋的方程和不变量，为相关理论提供新贡献。
Searching normal forms for real analytic submanifolds of C^n involves convergence problems. In 1983, J.K. Moser and S.M. Webster provided examples of real analytic surfaces in C^2 having an isolated hyperbolic (in the sense of E. Bishop) complex tangency, which are formally but not holomorphically normalizable (because of the presence of small divisors), even if the normal form is itself real analytic or algebraic. On the contrary, it appears that such a nonconvergence phenomenon does not appear for submanifolds of C^n whose CR dimension is locally constant, in view of recent results by S.M. B…
Fractal closures of geodesic planes in Hitchin manifolds
该研究通过构造 Hitchin 流形中的漂浮测地平面，证明了其闭包为分形，揭示了高秩无限体积情况下非紧类型局部对称空间的拓扑刚性反例。
We prove the validity of an inequality involving a mean of the area and the length of the boundary of immersed disks whose boundaries are homotopically non-trivial curves in an oriented compact manifold which possesses convex mean curvature boundary, positive escalar curvature and admits a map to $\mathbb{D}^2\times T^{n}$ with nonzero degree, where $\mathbb{D}^2$ is a disk and $T^n$ is an $n$-dimensional torus. We also prove a rigidity result for the equality case when the boundary is totally geodesic. This can be viewed as a partial generalization of a result due to Lucas Ambrózio in \cite{A…
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