Variationally complete actions on compact symmetric spaces

A unified approach to compact symmetric spaces of rank one

A relatively simple algebraic framework is given, in which all the compact symmetric spaces can be described and handled without distinguishing cases. We also give some applications and further results.

The First Eigenvalue of the Dirac Operator on Compact Outer Spin Symmetric Spaces

In two previous papers, we started a study of the first eigenvalue of the Dirac operator on compact spin symmetric spaces, providing, for symmetric spaces of "inner" type, a formula giving this first eigenvalue in terms of the algebraic data of the groups involved. We conclude here that study by giving the explicit expression of the first eigenvalue for "outer" compact spin symmetric spaces.

Polar actions on compact rank one symmetric spaces are taut

We prove that the orbits of a polar action of a compact Lie group on a compact rank one symmetric space are tautly embedded with respect to Z_2-coefficients.

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.

The First Eigenvalue of the Dirac Operator on Compact Spin Symmetric Spaces

We give a formula for the first eigenvalue of the Dirac operator acting on spinor fields of a spin compact irreducible symmetric space $G/K$.

The classifications of certain kind of isoparametric submanifolds in non-compact symmetric spaces

In this paper, we prove that full irreducible curvature-adapted isoparametric submanifolds of codimension greater than one in a symmetric space of non-compact type are principal orbits of Hermann actions on the symmetric spaces under certain conditions.

Biharmonic homogeneous hypersurfaces in compact symmetric spaces

In this paper, we study biharmonic hypersurfaces in Einstein manifolds. Then, we determine all the biharmonic hypersurfaces in irreducible symmetric spaces of compact type which are regular orbits of commutative Hermann actions of cohomogeneity one.

On the Funk transform on compact symmetric spaces

We prove that a function on an irreducible compact symmetric space M, which is not a sphere, is determined by its integrals over the shortest closed geodesics in M. We also prove a support theorem for the Funk transform on rank one symmetric spaces which are not spheres.

Horofunction Compactifications of Symmetric Spaces

We consider horofunction compactifications of symmetric spaces with respect to invariant Finsler metrics. We show that any (generalized) Satake compactification can be realized as a horofunction compactification with respect to a polyhedral Finsler metric.

Compact weakly symmetric spaces and spherical pairs

We prove that M. Kramer's classification of list of spherical pairs coincides with that for weakly symmetric spaces by examining the linear isotropy representation of the corresponding homogeneous space associated to each pair.

A formula for the First Eigenvalue of the Dirac Operator on Compact Spin Symmetric Spaces

Let $G/K$ be a simply connected spin compact inner irreducible symmetric space, endowed with the metric induced by the Killing form of $G$ sign-changed. We give a formula for the square of the first eigenvalue of the Dirac operator in terms of a root system of $G$. As an example of application, we give the list of the first eigenvalues for the spin compact irreducible symmetric spaces endowed with a quaternion-Kähler structure.

紧四元数凯勒对称空间的非紧全复子流形示例

An example of non-compact totally complex submanifolds of compact quaternionic Kähler symmetric spaces

本文构造了紧四元数凯勒对称空间（四元数射影空间除外）的非紧极大维全复子流形示例，分析其结构与群作用性质，并证明了相关紧子流形的存在性限制。

Vafa-Witten Estimates for Compact Symmetric Spaces

We give an optimal upper bound for the first eigenvalue of the untwisted Dirac operator on a compact symmetric space G/H with rk G-rk H\le 1 with respect to arbitrary Riemannian metrics. We also prove a rigidity statement.

Involutions of compact Riemannian 4-symmetric spaces

Let G/H be a compact 4-symmetric space of inner type such that the dimension of the center Z(H) of H is at most one. In this paper we shall classify involutions of G preserving H for the case where dim Z(H)=0, or H is a centralizer of a toral subgroup of G.

On compact splitting complex submanifolds of quotients of bounded symmetric domains

In the current article our primary objects of study are compact complex submanifolds of quotient manifolds of irreducible bounded symmetric domains by torsion free discrete lattices of automorphisms. We are interested in the characterization of the totally geodesic submanifolds among compact splitting complex submanifolds, i.e. under the assumption that the tangent sequence splits holomorphically over the submanifold.

Exploring the space of compact symmetric CMC surfaces

We map out the moduli space of Lawson symmetric constant mean curvature surfaces in the 3-sphere of genus $g>1$ by flowing numerically from Delaunay tori with even lobe count via the generalized Whitham flow.

Stability of Einstein metrics on symmetric spaces of compact type

We prove the linear stability with respect to the Einstein-Hilbert action of the symmetric spaces $\mathrm{SU}(n)$, $n\geq3$, and $E_6/F_4$. Combined with earlier results, this resolves the stability problem for irreducible symmetric spaces of compact type.

Four-dimensional compact Clifford-Klein forms of pseudo-Riemannian symmetric spaces with signature $(2, 2)$

We give a classification of irreducible four-dimensional symmetric spaces $G/H$ which admit compact Clifford-Klein forms, where $G$ is the transvection group of $G/H$. For this, we develop a method that applies to particular 1-connected solvable symmetric spaces. We also examine a `solvable analogue' of Kobayashi's conjecture for reductive groups and find an evidence that the reductive assumption in Kobayashi's conjecture is crucial.

紧致对称空间的切球丛上的不变接触度量结构

Invariant contact metric structures on tangent sphere bundles of compact symmetric spaces

本文刻画了紧致秩一对称空间，指出其球丛上的标准向量场关于某不变黎曼度量是Killing的，并确定了相关度量及接触结构；对秩≥2情形，构造了包含标准结构的不变接触度量结构族。

关于非紧秩一对称空间中Hopf对称集的PPW猜想

On the PPW Conjecture For Hopf-symmetric Sets In Non-compact Rank One Symmetric Space

本文证明了非紧秩一对称空间中Hopf对称集的PPW猜想，通过比较第二Dirichlet特征值与测地球体的特征值，扩展了之前在常曲率空间中的研究成果。

Ricci-flat Kähler metrics on tangent bundles of rank-one symmetric spaces of compact type

We give an explicit description of all complete $G$-invariant Ricci-flat Kähler metrics on the tangent bundle $T(G/K)\cong G^\bbC/K^\bbC$ of rank-one Riemannian symmetric spaces $G/K$ of compact type, in terms of associated vector-functions.

Polar actions on certain principal bundles over symmetric spaces of compact type

We study polar actions with horizontal sections on the total space of certain principal bundles $G/K\to G/H$ with base a symmetric space of compact type. We classify such actions up to orbit equivalence in many cases. In particular, we exhibit examples of hyperpolar actions with cohomogeneity greater than one on locally irreducible homogeneous spaces with nonnegative curvature which are not homeomorphic to symmetric spaces.

Hyperpolar isotropy actions on compact symmetric spaces

This paper has been withdrawn by the author due to a crucial error related with the consideration of regular points

Integral operators on the Oshima compactification of a Riemannian symmetric space of non-compact type. Microlocal analysis and kernel asymptotics

Let $\X\simeq G/K$ be a Riemannian symmetric space of non-compact type, $\widetilde \X$ its Oshima compactification, and $(π,\mathrm{C}(\widetilde \X))$ the regular representation of $G$ on $\widetilde \X$. We study integral operators on $\widetilde \X$ of the form $π(f)$, where $f$ is a rapidly falling function on $G$, and characterize them within the framework of pseudodifferential operators, describing the singular nature of their kernels. In particular, we consider the holomorphic semigroup generated by a strongly elliptic operator associated to the representation $π$, as well as its resol…

紧致对称空间上的测地线计数

Counting geodesics on compact symmetric spaces

本文通过焦轨道分解紧致对称空间指数映射的逆像，利用图、重数、Weyl群和格编码测地线的维数与连通分支，推广了紧致李群上的计数结果并给出已知结论的新证明。

Symmetric spaces of higher rank do not admit differentiable compactifications

Any nonpositively curved symmetric space admits a topological compactification, namely the Hadamard compactification. For rank one spaces, this topological compactification can be endowed with a differentiable structure such that the action of the isometry group is differentiable. Moreover, the restriction of the action on the boundary leads to a flat model for some geometry (conformal, CR or quaternionic CR depending of the space). One can ask whether such a differentiable compactification exists for higher rank spaces, hopefully leading to some knew geometry to explore. In this paper we answ…

Virtual immersions and minimal hypersurfaces in compact symmetric spaces

We show that closed, immersed, minimal hypersurfaces in a compact symmetric space satisfy a lower bound on the index plus nullity, which depends linearly on their first Betti number. Moreover, if either the minimal hypersurface satisfies a certain genericity condition, or if the ambient space is a product of two CROSSes, we improve this to a lower bound on the index alone, which is affine in the first Betti number. To prove these, we introduce a generalization of isometric immersions in Euclidean space. Compact symmetric spaces admit (and in fact are characterized by) such a structure with ske…

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。

Integral operators on the Oshima compactification of a Riemannian symmetric space of non-compact type. Regularized traces and characters

Consider a Riemannian symmetric space $X= G/K$ of non-compact type, where $G$ denotes a connected, real, semi-simple Lie group with finite center, and $K$ a maximal compact subgroup of $G$. Let $\widetilde X$ be its Oshima compactification, and $(π,\mathrm{C}(\widetilde X))$ the regular representation of $G$ on $\widetilde X$. In this paper, a regularized trace for the convolution operators $π(f)$ is defined, yielding a distribution on $G$ which can be interpreted as global character of $π$. In case that $f$ has compact support in a certain set of transversal elements, this distribution is a l…

Nonnegative curvature on stable bundles over compact rank one symmetric spaces

In this note we show that every (real or complex) vector bundle over a compact rank one symmetric space carries, after taking the Whitney sum with a trivial bundle of sufficiently large rank, a metric with nonnegative sectional curvature. We also examine the case of complex vector bundles over other manifolds, and give upper bounds for the rank of the trivial bundle that is necessary to add when the base is a sphere.

On the curvature of symmetric products of a compact Riemann surface

Let $X$ be a compact connected Riemann surface of genus at least two. The main theorem of arxiv:1010.1488 says that for any positive integer $n \leq 2({\rm genus}(X)-1)$, the symmetric product $S^n(X)$ does not admit any Kähler metric satisfying the condition that all the holomorphic bisectional curvatures are nonnegative. Our aim here is to give a very simple and direct proof of this result of Bökstedt and Romão.

非紧型单因子对称空间体积比较的研究

The volume comparison of symmetric spaces of non-compact type of rank 1

本文在标量曲率条件下，研究非紧型单因子对称空间的体积比较定理，并通过归一化Ricci-DeTurck流证明了刚性结果。

经典紧 Riemann 对称空间的极小子流形

Minimal Submanifolds of the Classical Compact Riemannian Symmetric Spaces

本研究面向经典紧 Riemann 对称空间，利用 Gudmundsson 与 Munn 的近期结果，提出借助特征函数构造极小子流形的方法。

Equifocal families in symmetric spaces of compact type

An equifocal submanifold M of a symmetric space N of compact type induces a foliation with singular leaves on N. In this paper we will show how to reconstruct the equifocal foliation starting from one of the singular leaves, the so-called focal manifolds. To be more concrete: The equifocal submanifold is equal to a partial tube B around the focal manifold and we will show how to construct B in this paper. Moreover, we will find a geometrical characterization of focal manifolds.

某些紧致洛伦兹局部对称空间的完备性

Completeness of certain compact Lorentzian locally symmetric spaces

该研究证明了满足局部de Rham-Wu分解中洛伦兹因子为Cahen-Wallach型或极大平坦因子为一维类时这两个条件之一的紧致洛伦兹局部对称空间是测地完备的，证明基于Mehidi与Zeghib的最新结果及Romero与Sánchez的早期结果。

Collapse of the mean curvature flow for isoparametric submanifolds in non-compact symmetric spaces

It is known that principal orbits of Hermann actions on a symmetric space of non-compact type are curvature-adapted isoparametric submanifolds having no focal point of non-Euclidean type on the ideal boundary of the ambient symmetric space. In this paper, we investigate the mean curvature flows for such a curvature-adapted isoparametric submanifold and its focal submanifold. Concretely the investigation is performed by investigating the mean curvature flows for the lift of the submanifold to an infinite dimensional pseudo-Hilbert space through a pseudo-Riemannian submersion.

紧致对称空间中极小超曲面的第一Betti数的指标估计

Index estimate by first Betti number of minimal hypersurfaces in compact symmetric spaces

本文证明紧致半单对称空间中不稳定闭极小超曲面的Morse指标被其第一Betti数的常数倍下界控制，方法为先前方法的自然推广，提供指标估计新途径。

Stability of Compact Symmetric Spaces

In this article we study the stability problem for the Einstein-Hilbert functional on compact symmetric spaces following and completing the seminal work of Koiso on the subject. We classify in detail the irreducible representations of simple Lie algebras with Casimir eigenvalue less than the Casimir eigenvalue of the adjoint representation, and use this information to prove the stability of the Einstein metrics on both the quaternionic and Cayley projective plane. Moreover we prove that the Einstein metrics on quaternionic Grassmannians different from projective spaces are unstable.

紧致局部对称洛伦兹流形的完备性

Completeness of compact Locally symmetric Lorentz manifolds

本文证明了所有紧致局部对称洛伦兹流形都是测地完备的，解决了该领域的猜想，并给出了四维情形的全面概述。

非紧型且高秩对称空间中的等参超曲面

Isoparametric hypersurfaces in symmetric spaces of non-compact type and higher rank

本文提出一种将子流形从欧几里得空间扩展到非紧型对称空间的新方法，构造出高秩对称空间中具有非严峻焦集的首批非齐次等参超曲面族。

Classification of isoparametric submanifolds admitting a reflective focal submanifold in symmetric spaces of non-compact type

In this paper, we assume that all isoparametric submanifolds have flat section. The main purpose of this paper is to prove that, if a full irreducible complete isoparametric submanifold of codimension greater than one in a symmetric space of non-compact type admits a reflective focal submanifold and if it of real analytic, then it is a principal orbit of a Hermann type action on the symmetric space. A hyperpolar action on a symmetric space of non-compact type admits a reflective singular orbit if and only if it is a Hermann type action. Hence is not extra the assumption that the isoparametric …

Polar foliations on symmetric spaces

We prove that a polar foliation of codimension at least three in an irreducible compact symmetric space is hyperpolar, unless the symmetric space has rank one. For reducible symmetric spaces of compact type, we derive decomposition results for polar foliations.

Invariant Ricci-flat Kähler metrics on tangent bundles of compact symmetric spaces

We give a description of all $G$-invariant Ricci-flat Kähler metrics on the canonical complexification of any compact Riemannian symmetric space $G/K$ of arbitrary rank, by using some special local $(1,0)$ vector fields on $T(G/K)$. As the simplest application, we obtain the explicit description of the set of all complete $\mathrm{SO}(3)$-invariant Ricci-flat Kähler metrics on $T{\mathbb S}^2$, which includes the well-known Eguchi-Hanson-Stenzel metrics and a new one-parameter family of metrics.

Scalar curvature estimates for compact symmetric spaces

We establish extremality of Riemannian metrics g with non-negative curvature operator on symmetric spaces M=G/K of compact type with rk(G)-rk(K)\le 1. Let g' be another metric with scalar curvature k', such that g'\ge g on 2-vectors. We show that k'\ge k everywhere on M implies k'=k. Under an additional condition on the Ricci curvature of g, k'\ge k even implies g'=g. We also study area-non-increasing spin maps onto such Riemannian manifolds.

DPW Potentials for Compact Symmetric CMC Surfaces in $\mathbb{S}^3$

Inspired by the work of Heller [12], we show that there exists a DPW potential for the Lawson surface $ξ_{k-1, l-1}$ from which it is possible to reconstruct the minimal immersion $f: ξ_{k-1, l-1} \to \mathbb{S}^3$ via the DPW method. Moreover, we extend the result to surfaces immersed in the 3-sphere with constant mean curvature which satisfy a certain symmetric condition.

基于复值特征函数的黎曼对称空间$SU(n)/SO(n)$、$Sp(n)/U(n)$、$SO(2n)/U(n)$、$SU(2n)/Sp(n)$的紧致极小子流形

Compact minimal submanifolds of the Riemannian symmetric spaces $SU(n)/SO(n)$, $Sp(n)/U(n)$, $SO(2n)/U(n)$, $SU(2n)/Sp(n)$ via complex-valued eigenfunctions

本研究利用复值特征函数，构造出四类经典黎曼对称空间的余维数为2的新型多维紧致极小子流形族，拓展了相关子流形研究的构造成果。

两个实旗流形在复旗流形中的交集，以及紧对称三元组的典范形式

The intersection of two real flag manifolds in a complex flag manifold, and the canonical form of a compact symmetric triad

该研究给出两个实旗流形在复旗流形中交集为离散的充要条件，证明离散交集为Weyl群轨道即对径集，并表明相关三元组可由典范的紧对称三元组构造。

A complete description of the antipodal set of most symmetric spaces of compact type

It is known that the antipodal set of a Riemannian symmetric space of compact type $G / K$ consists of a union of $K$-orbits. We determine the dimensions of these $K$-orbits of most irreducible symmetric spaces of compact type. The symmetric spaces we are not going to deal with are those with restricted root system $\mathfrak{a}_r$ and a non-trivial fundamental group, which is not isomorphic to $\mathbb{Z}_2$ or $\mathbb{Z}_{r+1}$. For example, we show that the antipodal sets of the Lie groups $Spin(2r+1)\:\: r\geq 5$, $E_8$ and $G_2$ consist only of one orbit which is of dimension $2r$, 128 a…

A duality between non-compact semisimple symmetric pairs and commutative compact semisimple symmetric triads and its general theory

The present paper investigates a natural generalization of the duality between Riemannian symmetric pairs of compact type and those of non-compact type à la É. Cartan. The main result of this paper is to construct an explicit description of a one-to-one correspondence between non-compact pseudo-Riemannian semisimple symmetric pairs and commutative compact semisimple symmetric triads, which is called the duality theorem. Further, we develop a general theory of the duality theorem.

Harmonic maps of finite uniton type into inner symmetric spaces

In this paper, we develop a loop group description of harmonic maps $\mathcal{F}: M \rightarrow G/K$ ``of finite uniton type", from a Riemann surface $M$ into inner symmetric spaces of compact or non-compact type. This develops work of Uhlenbeck, Segal, and Burstall-Guest to non-compact inner symmetric spaces. To be more concrete, we prove that every harmonic map of finite uniton type from any Riemann surface into any compact or non-compact inner symmetric space has a normalized potential taking values in some nilpotent Lie subalgebra, as well as a normalized frame with initial condition ident…

非紧黎曼对称空间SL_n(R)/SO(n), Sp(n,R)/U(n), SO*(2n)/U(n), SU*(2n)/Sp(n)的完整极小子流形 via 复值特征函数

Complete minimal submanifolds of the non-compact Riemannian symmetric spaces SL_n(R)/SO(n), Sp(n,R)/U(n), SO*(2n)/U(n), SU*(2n)/Sp(n) via complex-valued eigenfunctions

本文通过复值特征函数构造了非紧黎曼对称空间的二维完整极小子流形的新多维家族，展示了其核心方法与主要贡献。

On the kernel of the maximal flat Radon transform on symmetric spaces of compact type

Let $M$ be a Riemannian globally symmetric space of compact type, $M'$ its set of maximal flat totally geodesic tori, and $\mathrm{ad}(M)$ its adjoint space. We show that the kernel of the maximal flat Radon transform $τ:L^2(M) \rightarrow L^2(M')$ is precisely the orthogonal complement of the image of the pullback map $L^2(\mathrm{ad}(M))\rightarrow L^2(M)$. In particular, we show that the maximal flat Radon transform is injective if and only if $M$ coincides with its adjoint space.

A Cartan type identity for isoparametric hypersurfaces in symmetric spaces

In this paper, we obtain a Cartan type identity for curvature-adapted isoparametric hypersurfaces in symmetric spaces of compact type or non-compact type. This identity is a generalization of Cartan-D'Atri's identity for curvature-adapted(=amenable) isoparametric hypersurfaces in rank one symmetric spaces. Furthermore, by using the Cartan type identity, we show that certain kind of curvature-adapted isoparametric hypersurfaces in a symmetric space of non-compact type are principal orbits of Hermann actions.

紧致对称空间上的共形 Killing $2$-形式

Conformal Killing $2$-Forms on Compact Symmetric Spaces

本文证明了在单连通紧致对称空间中，存在非Killing的共形Killing 2-形式当且仅当该空间与球面、复射影空间或四元数射影空间同伦等价。