Keomkyo Seo
We prove that if an $n$-dimensional complete minimal submanifold $M$ in hyperbolic space has sufficiently small total scalar curvature then $M$ has only one end. We also prove that for such $M$ there exist no nontrivial $L^2$ harmonic 1-forms on $M$.
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In this paper, we study some basic geometric properties of pseudohermitian submanifolds of the Heisenberg groups. In particular, we obtain the uniqueness and existence theorems, and some rigidity theorems.
We find some integral formulas of Simons and Bochner type and use them to study biharmonic and biconservative submanifolds in space forms. We obtain rigidity results that in the biharmonic case represent partial answers to two well-known conjectures on such submanifolds in spheres.
Rigidity of Minimal Submanifolds in Spheres of Higher Codimension
该研究针对单位球面中高余维极小子流形的陈省身猜想，建立积分不等式并在第二基本形式平方模长挤压条件下获新刚性定理，推进了极小子流形刚性研究。
In this paper, we first introduce the concept of $ξ$-submanifold which is a natural generalization of self-shrinkers for the mean curvature flow and also an extension of $λ$-hypersurfaces to the higher codimension. Then, as the main result, we prove a rigidity theorem for Lagrangian $ξ$-submanifold in the complex $2$-plane $\bbc^2$.
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.
Rigidity of $k$-extremal submanifolds in a sphere
本文研究球面中$k(\ge 1)$-极值子流形的刚性，并在逐点和积分意义上的截面与Ricci曲率等不同条件下证明了多种夹挤定理。
In this note, we investigate the well-known Yau rigidity theorem for minimal submanifolds in spheres. Using the parameter method of Yau and the DDVV inequality verified by Lu, Ge and Tang, we prove that if $M$ is an $n$-dimensional oriented compact minimal submanifold in the unit sphere $S^{n+p}(1)$, and if $K_{M}\geq\frac{sgn(p-1)p}{2(p+1)},$ then $M$ is either a totally geodesic sphere, the standard immersion of the product of two spheres, or the Veronese surface in $S^4(1)$. Here $sgn(\cdot)$ is the standard sign function. We also extend the rigidity theorem above to the case where $M$ is a…
We show that an $n$-dimensional surface whose entropy is close to that of an $n$-dimensional plane is close in Hausdorff distance to some $n$-dimensional plane at every scale. Moreover we show that self-expanders of low entropy converge in the Hausdorff sense to their asymptotic cones.
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 $\mathbb{H}^{4}$.
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。
Let $F^{n+p}(c)$ be an $(n+p)$-dimensional simply connected space form with nonnegative constant curvature $c$. We prove that if $M^n(n\geq4)$ is a compact submanifold in $F^{n+p}(c)$, and if $Ric_M>(n-2)(c+H^2),$ where $H$ is the mean curvature of $M$, then $M$ is homeomorphic to a sphere. We also show that the pinching condition above is sharp. Moreover, we obtain a new differentiable sphere theorem for submanifolds with positive Ricci curvature.
Geometric and topological rigidity of pinched submanifolds
针对空间形式中满足第二基本形式长度与平均曲率挤压条件的任意余维数紧致子流形，证明其要么使中间维度同调消失，要么确定子流形至全等，结果精确且无额外平均曲率假设地推广了前人成果。
We study isoparametric submanifolds of rank at least two in a separable Hilbert space, which are known to be homogeneous by a result of Heintze and Liu, and associate to such a submanifold M and a point x in M a canonical homogeneous structure (a certain bilinear map on the tangent space). We prove that the homogeneous structure together with the second fundamental form encodes all the information about M, and deduce from this the rigidity result that M is completely determined by the second fundamental form and its covariant derivative, thereby making such submanifolds accessible to classific…
We show that among the Euclidean submanifolds with codimension two the ones of rank two that are parabolic but nonruled are isometrically rigid. This generalizes the result in [10] that these submanifolds are genuinely rigid. In addition, we give a parametric classifications of all parabolic submanifolds.
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…
The rigidity of minimal Legendrian submanifolds in the Euclidean spheres via eigenvalues of fundamental matrices
本文利用基本矩阵特征值与 Lu 不等式，建立单位球中各维紧致极小 Legendrian 子流形的最优夹挤与刚性定理，并刻画 Calabi 环面。
We propose a unified computational framework for the problem of deformation and rigidity of submanifolds in a homogeneous space under geometric constraint. A notion of 1-rigidity of a submanifold under admissible deformations is introduced. It measures how a deformation deviates from a one parameter family of motions up to 1st order. We implement this method to rigidity of CR submanifolds in spheres. A class of submanifolds called Bochner rigid submanifolds are shown to be 1-rigid under type preserving CR deformations. This 1-rigidity is then extended to a local rigidity, which roughly states …
Rigidity Results for Compact Submanifolds with Pinched Ricci Curvature in Euclidean and Spherical Space Forms
本文研究了在欧几里得和球面空间形式中具有受限曲率的紧致子流形，证明其要么是爱因斯坦克利福德 тор斯，要么是拓扑球体，或具有部分同调群消失，扩展了多项经典刚性结果。
Geometric and topological rigidity of pinched submanifolds in Riemannian manifolds
研究Riemannian流形中满足第二基本形式范数与平均曲率的严格pinching条件的紧致子流形的刚性，揭示其几何与拓扑限制，扩展了球面定理等已知结果。
In this paper, we study Lagrangian submanifolds of the homogeneous nearly Kähler $6$-dimensional unit sphere $\mathbb{S}^6(1)$. As the main result, we derive a Simons' type integral inequality in terms of the second fundamental form for compact Lagrangian submanifolds of $\mathbb{S}^6(1)$. Moreover, we show that the equality sign occurs if and only if the Lagrangian submanifold is either the totally geodesic $\mathbb{S}^3(1)$ or the Dillen-Verstraelen-Vrancken's Berger sphere $S^3$ discribed in J Math Soc Japan, 42: 565-584, 1990.
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…
Geometric and topological rigidity of pinched submanifolds II
本文研究空间形式中满足夹条件的四维紧致子流形的几何与拓扑刚性，利用四维几何、非负各向同性曲率及Bochner技巧，得到精确结果并推广先前工作，无需额外假设。
In this paper, we obtain a rigidity theorem for Lagrangian submanifolds of $C^n$ and $CP^n$ with conformal Maslov form.
Topological and geometric rigidity of nonnegatively curved submanifolds
研究非负曲率空间形式中满足特定条件的紧致子流形，通过特定方法探究其拓扑与几何，证明条件对其拓扑或几何有强约束并给出精确性例子。
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.
Let $M$ be an $n(\geq 4)$-dimensional compact submanifold in the simply connected space form $F^{n+p}(c)$ with constant curvature $c\geq 0$, where $H$ is the mean curvature of $M$. We verify that if the scalar curvature of $M$ satisfies $R>n(n-2)(c+H^2)$, and if $Ric_M\geq (n-2-\frac{2σ_n}{2n-σ_n})(c+H^2)$, then $M$ is homeomorphic to a sphere. Here $σ_n=sgn(n-4)((-1)^n+3)$, and $sgn(\cdot)$ is the standard sign function. This improves our previous sphere theorem \cite{XG2}. It should be emphasized that our pinching conditions above are optimal. We also obtain some new topological sphere theor…
Minimal homogeneous submanifolds of complex hyperbolic spaces
本文研究复超球面的极小同构子流形的分类，采用同构性条件和几何分析方法，揭示了其结构特征和分类结果。
We classify homogeneous CR submanifolds in complex hyperbolic spaces arising as orbits of a subgroup of the solvable part of the Iwasawa decomposition of the isometry group of the ambient space.
Rigidity Result for Four Dimensional Willmore Submanifolds with Boundary
本文研究了带边界的四维Willmore能量临界点的刚性，通过建立曲率估计，证明了满足完全测地边界条件的四维Willmore子流形是脐性的。
A rigidity result for coisotropic submanifolds in contact geometry
本文研究接触流形中紧致正则余迷向子流形的余迷向形变，证明了在特征叶层微分同胚的邻近子流形中其具有刚性，并结合经典叶层刚性结果推广至所有邻近子流形。
On minimal homogeneous submanifolds of the hyperbolic space up to codimension two
本文研究双曲空间中余维数不超过二的齐次极小子流形，证明维数 n≥5 时此类子流形必为全测地子流形。
A topological sphere theorem for submanifolds of the hyperbolic space
本文证明，双曲空间中满足里奇曲率下界（仅依赖平均曲率向量长度）的完备子流形必为拓扑球面。
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 $Ric_{M}>(n-2)(c+H^2),$ …
Unlike Legendrian submanifolds, the deformation problem of coisotropic submanifolds can be obstructed. Starting from this observation, we single out in the contact setting the special class of integral coisotropic submanifolds as the direct generalization of Legendrian submanifolds for what concerns deformation and moduli theory. Indeed, being integral coisotropic is proved to be a rigid condition, and moreover the integral coisotropic deformation problem is unobstructed with discrete moduli space.
Rigidity of Lagrangian submanifolds with conformal Maslov form and Legendrian capillary boundary
本文分类了单位球中具有共形Maslov形式和勒让德毛细边界的拉格朗日浸入球，证明其像为赤道圆盘或Whitney球，并证实了二维情形下的一个猜想。
In this paper, we prove the nonexistence of $L^2$ harmonic 1-forms on a complete super stable minimal submanifold $M$ in hyperbolic space under the assumption that the first eigenvalue $λ_1 (M)$ for the Laplace operator on $M$ is bounded below by $(2n-1)(n-1)$. Moreover, we provide sufficient conditions for minimal submanifolds in hyperbolic space to be super stable.
In this paper, we consider the rigidity for an $n(\geq 4)$-dimensional submanfolds $M^n$ with parallel mean curvature in the space form ${\mathbb M}^{n+p}_c$ when the integral Ricci curvature of $M$ has some bound. We prove that, if $c+H^2>0$ and $\|\mathrm{Ric}_{-}^λ\|_{n/2}< ε(n,c, λ, H)$ for $λ$ satisfying $ \frac{n-2}{n-1} (c+H^2) < λ\le c+H^2$, then $M$ is the totally umbilical sphere $\mathbb{S}^n(\tfrac{1}{\sqrt{c+H^2}})$. Here $H$ is the norm of the parallel mean curvature of $M$, and $ε(n,c,λ, H)$ is a positive constant depending only on $n, c,λ$ and $H$. This extends some of the earl…
This paper explores the topology of monotone Lagrangian submanifolds $L$ inside a symplectic manifold $M$ by exploiting the relationships between the quantum homology of $M$ and various quantum structures associated to the Lagrangian $L$.
We prove a rigidity theorem in Poisson geometry around compact Poisson submanifolds, using the Nash-Moser fast convergence method. In the case of one-point submanifolds (fixed points), this immediately implies a stronger version of Conn's linearization theorem, also proving that Conn's theorem is, indeed, just a manifestation of a rigidity phenomenon; similarly, in the case of arbitrary symplectic leaves, it gives a stronger version of the local normal form theorem; another interesting case corresponds to spheres inside duals of compact semisimple Lie algebras, our result can be used to fully …
On the essential spectra of submanifolds in the hyperbolic space
该研究探讨双曲空间中子流形渐近几何与理想边界附近正则性的关联，分析极小子流形与理想边界正交的假设，计算渐近极小子流形上拉普拉斯算子的本质谱，补充了相关领域的谱分析结果。
In this work, we establish new rigidity results for the Maslov class of Lagrangian submanifolds in large classes of closed and convex symplectic manifolds. Our main result establishes upper bounds for the minimal Maslov number of displaceable Lagrangian submanifolds which are product manifolds whose factors each admit a metric of negative sectional curvature. Such Lagrangian submanifolds exist in every symplectic manifold of dimension greater than six or equal to four. The proof utilizes the relations between closed geodesics on the Lagrangian, the periodic orbits of geometric Hamiltonian flow…
We investigate complete minimal submanifolds $f\colon M^3\to\Hy^n$ in hyperbolic space with index of relative nullity at least one at any point. The case when the ambient space is either the Euclidean space or the round sphere was already studied in \cite{dksv1} and \cite{dksv2}, respectively. If the scalar curvature is bounded from below we conclude that the submanifold has to be either totally geodesic or a generalized cone over a complete minimal surface lying in an equidistant submanifold of $\Hy^n$.
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。
Mean Curvature Flow for Isoparametric Submanifolds in Hyperbolic Spaces
本文研究双曲空间中等参子流形的均曲率流，证明其总是存在古老解并研究其极限行为。
Ricci solitons as submanifolds of complex hyperbolic spaces
本文针对对称空间为复双曲空间的情况，对具有Ricci孤立子诱导度量的、非紧型对称空间Iwasawa分解可解部分的李子群开展分类与几何性质分析。
We extend the concept of genuine rigidity of submanifolds by allowing mild singularities, mainly to obtain new global rigidity results and unify the known ones. As one of the consequences, we simultaneously extend and unify Sacksteder and Dajczer-Gromoll theorems by showing that any compact $n$-dimensional submanifold of ${\mathbb R}^{n+p}$ is singularly genuinely rigid in ${\mathbb R}^{n+q}$, for any $q < \min\{5,n\} - p$. Unexpectedly, the singular theory becomes much simpler and natural than the regular one, even though all technical codimension assumptions, needed in the regular case, are …
In this paper we construct new examples of minimal Lagrangian submanifolds in the complex hyperbolic space with large symmetry groups, obtaining three 1-parameter families with cohomegeneity one. We characterize them as the only minimal Lagrangian submanifolds in CH^n foliated by umbilical hypersurfaces of Lagrangian subspaces RH^n of CH^n. Several suitable generalizations of the above construction allow us to get new families of minimal Lagrangian submanifolds in CH^n from curves in CH^1 and (n-1)-dimensional minimal Lagrangian submanifolds of the complex space forms CP^n-1, CH^n-1 and C^n-1.…
In this note, we study deformations of quaternionic hyperbolic lattices in larger quaternionic hyperbolic spaces and prove local rigidity results. On the other hand, surface groups are shown to be more flexible in quaternionic hyperbolic plane than in complex hyperbolic plane.
Hyperbolic solitons on trans-Sasakian space forms and its submanifolds
引入双曲$*-$Ricci孤子和双曲Ricci-Yamabe孤子概念，研究跨Sasakian空间形式及其子流形上孤子的性质。
Minimal Submanifolds of The Complex and Quaternionic Projective and Hyperbolic Spaces $\cn P^{2n-1}$, $\hn P^{2n-1}$, $\cn H^{2n-1}$, $\hn H^{2n-1}$ via Harmonic Morphisms
本文利用复值谐波映射构造了复和四元数投影双曲空间中的非全纯、完整极小子流形，其共轭空间也包含极小子流形，所有构造的极小子流形均为二阶余维。
A rigidity result for weakly asymptotically hyperbolic manifolds with lower bounds on Ricci curvature is proved without assuming that the manifolds are spin. The argument makes use of a quasi-local mass characterization of Euclidean balls from \cite{Miao} \cite{S_T} and eigenfunction compactification ideas from \cite{Qing}.
We present several local and global results on isometric immersions of Kaehler manifolds $M^{2n}$ into hyperbolic space $\Hy^{2n+p}$. For instance, a classification is given in the case of dimension $n\geq 4$ and codimension $p=2$. Moreover, as corollaries of general results, we conclude that there are no isometric immersion in codimension $p\leq n-2$ if the Kaehler manifold is of dimension $n\geq 4$ and either has a point of positive holomorphic sectional curvature or is compact.
When $X=Γ\backslash \H^n$ is a real hyperbolic manifold, it is already known that if the critical exponent is small enough then some cohomology spaces and some spaces of $L^2$ harmonic forms vanish. In this paper, we show rigidity results in the borderline case of these vanishing results.
Dihedral rigidity for submanifolds of warped product manifolds
本文研究扭曲乘积流形中带多面体边界的余维零子流形，证明了放宽面方向及叶曲率、扭曲函数条件下的二面角极值与刚性定理。
There are several well-known characterizations of the sphere as a regular surface in the Euclidean space. By means of a purely synthetic technique, we get a rigidity result for the sphere without any curvature conditions, nor completeness or compactness. As well as a dual result for the hyperbolic plane, the spacelike sphere in the Minkowski space.
Relative Liouville Rigidity for Lagrangian Self-Similar Submanifolds with Legendrian Boundary
研究单位球中具有勒让德边界的紧致拉格朗日子流形，证明在精确性或接触角余弦符号假设下，自相似子流形必为赤道n-盘，并将二维极小刚性定理推广到高维和拉格朗日自相似情形。
The rigidity of marked length spectrum for closed hyperbolic surfaces due to Fricke-Klein [7] has been the motivation of many different rigidity results, specially for manifolds of negative curvature. From the works of Vigneras [18], Sunada [17] and many other authors this result is far from being true for the unmarked length spectrum. The purpose of this paper is to introduce a closely a related unmarked spectrum, the length-angle spectrum, and show that it determines the surface uniquely.
In this survey, we will focus on the mean curvature flow theory with sphere theorems, and discuss the recent developments on the convergence theorems for the mean curvature flow of arbitrary codimension inspired by the Yau rigidity theory of submanifolds. Several new differentiable sphere theorems for submanifolds are obtained as consequences of the convergence theorems for the mean curvature flow. It should be emphasized that Theorem 4.1 is an optimal convergence theorem for the mean curvature flow of arbitrary codimension, which implies the first optimal differentiable sphere theorem for sub…
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…
Kaehler submanifolds of the real hyperbolic space
针对实双曲空间中低余维Kaehler子流形局部分类的开放问题，本文将球面子流形的相关分类结论推广到等距浸入双曲空间的Kaehler流形，同时推广了Dajczer与Vlachos的多项结果。
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