Thomas A. Ivey, Patrick J. Ryan
Using the methods of moving frames, we study real hypersurfaces in complex projective space CP^2 and complex hyperbolic space CH^2 whose structure Jacobi operator has various special properties. Our results complement work of several other authors who worked on such hypersurfaces in CP^n and CH^n for n>2.
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We classify real hypersurfaces in CP^2and CH^2 equipped with pseudo-parallel structure Jacobi operator.
We prove that there does not exist any real hypersurface in complex Grassmannians of rank two with semi-parallel structure Jacobi operator. With this result, the nonexistence of real hypersurface in complex Grassmannians of rank two with recurrent structure Jacobi operator is proved.
We prove the non-existence of real hypersurfaces in CP^2 and CH^2 whose structure Jacobi operator is Lie D-parallel.
In this paper, we introduce a notion of quadratic Killing structure Jacobi operator (simply, Killing structure Jacobi operator) and its geometric meaning for real hypersurfaces in the complex two-plane Grassmannians. In addition, we give a classification theorem for Hopf real hypersurfaces with quadratic Killing structure Jacobi operator in complex two-plane Grassmannians.
We study three dimensional real hypersurfaces in CP^2 and CH^2 equipped with $xi$-parallel structure Jacobi operator. We prove that they are Hopf hypersurfaces and if additional $α\neq0$, we classify them.
We introduce the notion of Reeb parallel structure Jacobi operator for real hypersurfaces in the complex hyperbolic quadric ${Q^*}^m=SO^0_{2,m}/SO_2 SO_m$, $m \geq 3$, and give a classification theory for real hypersurfaces in ${{Q^*}^m}$, $m \geq 3$, with Reeb parallel structure Jacobi operator.
First we introduce the notion of parallel structure Jacobi operator for real hypersurfaces in the complex quadric $Q^m = SO_{m+2}/SO_mSO_2$ . Next we give a complete classification of real hypersurfaces in $Q^m = SO_{m+2}/SO_mSO_2$ with parallel structure Jacobi operator.
The aim of the present paper is the study of real hypersurfaces equipped with the condition $ϕl = l ϕ$, $l = R(., ξ, ξ)$.
The objective of the present paper is to prove the non-existence of real hypersurface with pseudo-parallel normal Jacobi operator in complex two-plane Grassmannians. As a corollary, we show that there does not exist any real hypersurface with semi-parallel or recurrent normal Jacobi operator in complex two-plane Grassmannians. This answers a question considered in [Monatsh Math, 172 (2013), 167-178] in negative.
In this paper we prove some classification theorems of real hypersur- faces in Mn(c) satisfying certain conditions on the covariant derivative of the structure Jacobi operator. We also prove the non-existence of real hypersurfaces with Codazzi type structure Jacobi operator in Mn(c).
The aim of the present paper is the study of some classes of real hypersurfaces equipped with the condition ϕl = l ϕ, (l = R(., ξ, ξ))
In \cite{S 2017}, Suh gave a non-existence theorem for Hopf real hypersurfaces in the complex quadric with parallel normal Jacobi operator. Motivated by this result, in this paper, we introduce some generalized conditions named $\mathcal C$-parallel or Reeb parallel normal Jacobi operators. By using such weaker parallelisms of normal Jacobi operator, first we can assert a non-existence theorem of Hopf real hypersurfaces with $\mathcal C$-parallel normal Jacobi operator in the complex quadric $Q^{m}$, $m \geq 3$. Next, we prove that a Hopf real hypersurface has Reeb parallel normal Jacobi opera…
In this paper, we study the first eigenvalue of Jacobi operator on an $n$-dimensional non-totally umbilical compact hypersurface with constant mean curvature $H$ in the unit sphere $S^{n+1}(1)$. We give an optimal upper bound for the first eigenvalue of Jacobi operator, which only depends on the mean curvature $H$ and the dimension $n$.
In this paper, first, we investigate the commuting property between the normal Jacobi operator~${\bar R}_N$ and the structure Jacobi operator~$R_ξ$ for Hopf real hypersurfaces in the complex quadric~$Q^m = SO_{m+2}/SO_mSO_2$, $m \geq 3$, which is defined by ${\bar R}_N R_ξ = R_ξ{\bar R}_N$. Moreover, a new characterization of Hopf real hypersurfaces with $\mathfrak A$-principal singular normal vector field in the complex quadric~$Q^{m}$ is obtained. By virtue of this result, we can give a remarkable classification of Hopf real hypersurfaces in the complex quadric~$Q^{m}$ with commuting Jacobi …
In this paper, we have considered a new commuting condition, that is, $(R_ξϕ) S = S (R_ξϕ)$ \big(resp. $(\Bar{R}_Nϕ) S = S (\Bar{R}_Nϕ$)\big) between the restricted Jacobi operator~$R_ξϕ$ (resp. $\Bar{R}_Nϕ$), and the Ricci tensor $S$ for real hypersurfaces $M$ in $G_2({\mathbb C}^{m+2})$. In terms of this condition we give a complete classification for Hopf hypersurfaces $M$ in $G_2({\mathbb C}^{m+2})$.
In this paper, we introduce a new commuting condition between the structure Jacobi operator and symmetric (1,1)-type tensor field $T$, that is, $R_ξϕT=TR_ξϕ$, where $T=A$ or $T=S$ for Hopf hypersurfaces in complex hyperbolic two-plane Grassmannians. By using simultaneous diagonalzation for commuting symmetric operators, we give a complete classification of real hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting condition respectively.
In this work we characterize certain immersed closed hypersurfaces of some ambient manifolds via the second eigenvalue of the Jacobi operator. First, we characterize the Clifford torus as the surface which maximizes the second eigenvalue of the Jacobi operator among all closed immersed orientable surfaces of $\mathbb S^3$ with genus bigger than zero. After, we characterize the slices of the warped product $I\times_h\mathbb S^n$, under a suitable hypothesis on the warping function $h:I\subset\mathbb R\to\mathbb R$, as the only hypersurfaces which saturate a certain integral inequality involving…
It is proved the non-existence of Hopf hypersurfaces in $G_{2}({\Bbb C}^{m+2})$, $m \geq 3$, whose normal Jacobi operator is semi-parallel, if the principal curvature of the Reeb vector field is non-vanishing and the component of the Reeb vector field in the maximal quaternionic subbundle ${\frak D}$ or its orthogonal complement ${\frak D}^{\bot}$ is invariant by the shape operator.
Let $M$ be a real hypersurface of a complex space form with almost contact metric structure $(ϕ, ξ, η, g)$. In this paper, we study real hypersurfaces in a complex space form whose structure Jacobi operator $R_ξ=R(\cdot,ξ)ξ$ is $ξ$-parallel. In particular, we prove that the condition $\nabla_ξ R_ξ=0$ characterizes the homogeneous real hypersurfaces of type $A$ in a complex projective space or a complex hyperbolic space when $R_ξ ϕS=S ϕR_ξ$ holds on $M$, where $S$ denotes the Ricci tensor of type (1,1) on $M$.
In this paper, we first introduce the full express of the Riemannian curvature tensor of a real hypersurface $M$ in complex quadric $Q^{m}$ from the equation of Gauss. Next we derive a formula for the structure Jacobi operator $R_ξ$ and its derivative under the Levi-Civita connection of $M$. We give a complete classification of Hopf real hypersurfaces with Reeb parallel structure Jacobi operator, $\nabla_ξR_ξ =0$, in the complex quadric $Q^{m}$, $m \geq 3$.
This paper presents two results conserning real hypersurfaces in CP^{2} and CH^{2}. More precisely, it is proved that real hypersurfaces equipped with structure Jacobi operator satisfying condition $\mathcal{L}_{X}l=\nabla_{X}l$, where \emph{X} is a vector field orthogonal to structure vector field $ξ$, do not exist. Additional real hypersurfaces equipped with shape operator \emph{A} satisfying relation $\mathcal{L}_{X}A=\nabla_{X}A$, where \emph{X} is a vector field orthogonal to $ξ$, do not exist.
In this paper we study real hypersurfaces in the complex quadric space $Q^m$ whose structure Jacobi operator commutes with their structure tensor field. We show that the Reeb curvature $α$ of such hypersurfaces is constant and if $α$ is non-zero then the hypersurface is a tube around a totally geodesic submanifold $\mathbb{C} P^k \subset Q^m$, where $m=2k$. We also consider Reeb flat hypersurfaces, namely, when the Reeb curvature is zero. We show that the tube around $\mathbb{C} P^k \subset Q^m$ ($m=2k$), with radius $\fracπ{4}$ is the only Reeb flat Hopf hypersurface with commuting Ricci tens…
Using generalized Tanaka-Webster connection, we considered a real hypersurface $M$ in a complex two-plane Grassmannian $G_2({\mathbb C}^{m+2})$ when the GTW Reeb Lie derivative of the structure Jacobi operator coincides with the Reeb Lie derivative. Next using the method of simultaneous diagonalization, we prove a complete classification for a real hypersurface in $G_2({\mathbb C}^{m+2})$ satisfying such a condition. In this case, we have proved that $M$ is an open part of a tube around a totally geodesic $G_2({\mathbb C}^{m+1})$ in $G_2({\mathbb C}^{m+2})$.
Given a closed Riemannian manifold of dimenion less than eight, we prove a compactness result for the space of closed, embedded minimal hypersurfaces satisfying a volume bound and a uniform lower bound on the first eigenvalue of the stability operator. When the latter assumption is replaced by a uniform lower bound on the p-th Jacobi eigenvalue for p greater or equal than 2 one gains strong convergence to a smooth limit submanifold away from at most p-1 points.
Let $M$ be a real hypersurface in complex projective space. The almost contact metric structure on $M$ allows us to consider, for any nonnull real number $k$, the corresponding $k$-th generalized Tanaka-Webster connection on $M$ and, associated to it, a differential operator of first order of Lie type. Considering such a differential operator and Lie derivative we define, from the structure Jacobi operator $R_ξ$ on $M$ a tensor field of type (1,2), $R_{ξ_T}^{(k)}$. We obtain some classifications of real hypersurfaces for which $R_{ξ_T}^{(k)}$ is either symmetric or skew symmetric.
The Jacobi operator of some special minimal hypersurfaces
本文研究渐近于Lawson锥$C_{m,n}$的$\u211d^m\times\u211d^n$中极小超曲面的稳定性与非退化性，分$m+n\geq8$和$m+n\leq7$两种情形给出严格稳定、非退化及无穷Morse指数的结论。
Let $x:M\to\mathbb{S}^{n+1}(1)$ be an n-dimensional compact hypersurface with constant scalar curvature $n(n-1)r,~r\geq 1$, in a unit sphere $\mathbb{S}^{n+1}(1),~n\geq 5$. We know that such hypersurfaces can be characterized as critical points for a variational problem of the integral $\int_MH dv$ of the mean curvature $H$. In this paper, we derive an optimal upper bound for the second eigenvalue of the Jacobi operator $J_s$ of $M$. Moreover, when $r>1$, the bound is attained if and only if $M$ is totally umbilical and non-totally geodesic, when $r=1$, the bound is attained if $M$ is the Riem…
We show how some of well-known recurrent operators such as recurrent curvature operator, recurrent Ricci operator, recurrent Jacobi operator, recurrent shape and Weyl operators have the significant role for biharmonic hypersurfaces to be minimal in the Euclidean space.
Fredholm properties of the jacobi Operator of minimal conical hypersurfaces
本文研究了极小锥面的Jacobi算子的Fredholm性质，证明了在一定条件下该算子的可解性，并讨论了应用实例。
First Eigenvalue of Jacobi operator and Rigidity Results for Constant Mean Curvature Hypersurfaces
本文研究了常均曲率超曲面的Jacobi算子特征值及其刚性结果，推导了面积和边界长度的刚性定理，并探讨了Yamabe不变量在高维中的应用。
Given a compact Riemannian manifold with boundary, we prove that the space of embedded, which may be improper, free boundary minimal hypersurfaces with uniform area and Morse index upper bound is compact in the sense of smoothly graphical convergence away from finitely many points. We show that the limit of a sequence of such hypersurfaces always inherits a non-trivial Jacobi field when it has multiplicity one. In a forthcoming paper, we will construct Jacobi fields when the convergence has higher multiplicity.
Let $M\subset \mathbb{S}^{n+1}\subset\mathbb{R}^{n+2}$ be a compact cmc rotational hypersurface of the $(n+1)$-dimensional Euclidean unit sphere. Denote by $|A|^2$ the square of the norm of the second fundamental form and $J(f)=-Δf-nf-|A|^2f$ the stability or Jacobi operator. In this paper we compute the spectra of their Laplace and Jacobi operators in terms of eigenvalues of second order Hill's equations. For the minimal rotational examples, we prove that the stability index --the numbers of negative eigenvalues of the Jacobi operator counted with multiplicity -- is greater than $3 n+4$ and w…
In this paper, we introduce new notions of semi-parallel shape operators and structure Jacobi operators in complex two-plane Grassmannians $G_2({\mathbb C}^{m+2})$. By using such a semi-parallel condition, we give a complete classification of Hopf hypersurfaces in $G_2({\mathbb C}^{m+2})$.
In this study, we obtain a spinorial Gauss formula for a lightlike hypersurface in Lorentzian manifold with 4-dimension. Then, we take into account the changes caused by degenerate metric on hypersurface and investigate Dirac operator for lightlike hypersurface. Later, we establish the relation between Dirac operators and Riemannian curvatures of manifold and hypersurface.
CR-hypersurfaces of conformal Kenmotsu space form satisfying certain shape operator conditions
We consider the Jacobi operator, defined on a closed oriented hypersurfaces immersed in the Euclidean space with the same volume of the unit sphere. We show a local generalization for the classical result of the Willmore functional for the Euclidean sphere. As a consequence, we prove that the first eigenvalue of the Jacobi operator in the Euclidean sphere is a local maximum and this result is a global one in the closed oriented surfaces space of $\mathbb{R}^3$ and genus zero.
We study deformations of free boundary constant mean curvature (CMC) hypersurfaces whose Jacobi operator is degenerate due to symmetries of the ambient space. The value of the mean curvature and the ambient metric are allowed to vary simultaneously, provided that the infinitesimal ambient symmetries change smoothly. We discuss applications to free boundary CMC disks and Delaunay annuli in the unit ball of a space form.
In this paper, we study biconservative hypersurfaces in the four dimensional Minkowski space $\mathbb E^4_1$. We give the complete explicit classification of biconservative hypersurfaces with diagonalizable shape operator in $\mathbb E^4_1$.
In this paper, we obtain some properties of biconservative Lorentz hypersurface $M_{1}^{n}$ in $E_{1}^{n+1}$ having shape operator with complex eigen values. We prove that every biconservative Lorentz hypersurface $M_{1}^{n}$ in $E_{1}^{n+1}$ whose shape operator has complex eigen values with at most five distinct principal curvatures has constant mean curvature. Also, we investigate such type of hypersurface with constant length of second fundamental form having six distinct principal curvatures.
We complete the classification of ruled real hypersurfaces with shape operator of constant norm in nonflat complex space forms by showing the existence of a unique inhomogeneous example in the complex hyperbolic space.
A construction of curvature-adapted hypersurfaces in the product of symmetric spaces
本文在对称空间乘积中构造曲率适应超曲面，提供大量实例并计算其形状算子与法向 Jacobi 算子的特征值。
In this paper, we investigate the preservability of the curvature-adaptedness along the mean curvature flow starting from a compact curvature-adapted hypersurface in locally symmetric spaces, where the curvature-adaptedness means that the shape operator and the normal Jacobi operator of the hypersurface commute.
We study Lorentz hypersurfaces $M_{1}^{n}$ in $E_{1}^{n+1}$ satisfying $\triangle \vec {H}= α\vec {H}$ with non diagonal shape operator, having complex eigenvalues. We prove that every such Lorentz hypersurface in $E_{1}^{n+1}$ having at most five distinct principal curvatures has constant mean curvature.
We generalize the transgression formula for the eta form of Bismut, Cheeger and Berline, Getzler, Vergne for vertical Dirac operators on a fibre bundle with odd dimensional fibres where the Dirac operators have locally at most one eigenvalue of multiplicity one crossing zero transversally.
Stark hypersurfaces are a special class of austere hypersurface in ${\mathbb C}P^n$ where the shape operator is compatible with the $CR$-structure. In this paper, the possible shape operators for stark hypersurfaces are completely determined, and stark hypersurfaces in ${\mathbb C}P^2$ are constructed as integrals of a Frobenius exterior differential system.
We characterize Riemannian manifolds of constant sectional curvature in terms of commutation properties of their Jacobi operators.
We deal with hypersurfaces in the framework of the $n$-dimensional relative differential geometry. We consider a hypersurface $\varPhi$ of $\mathbb{R}^{n+1}$ with position vector field $\mathbf{x}$, which is relatively normalized by a relative normalization $\mathbf{y}$. Then $\mathbf{y}$ is also a relative normalization of every member of the one-parameter family $\mathcal{F}$ of hypersurfaces $\varPhi_μ$ with position vector field $$\mathbf{x}_μ= \mathbf{x} + μ\, \mathbf{y},$$ where $μ$ is a real constant. We call every hypersurface $\varPhi_μ\in \mathcal{F}$ relatively parallel to $\varPhi$…
Let $M$ be a globally hyperbolic manifold with complete spacelike Cauchy hypersurface $Σ\subset M$. Building on past and recent works of Bär and Strohmaier, we extend their Fredholm result of the Atiyah-Singer Dirac operator on compact Lorentzian spaces to the case, where $M$ is diffeomorphic to a product of $Σ$ with a compact time intervall and the hypersurface is a Galois covering with respect to a group $Γ$. We follow the first approach of both authors in this extended setting, where a well-posedness result of the Cauchy problem for the Dirac operator on non-compact manifolds is needed in p…
We extend the potential theory on almost minimzers from Part 1. We introduce so-called Hardy structures to study many classical operators using the tools from part 1. Furthermore, we show that for a naturally defined operator L, minimal growth of positive solutions of Lw = 0 towards the singular set is a stable property. It persists under perturbations or blow-ups of the underlying spaces. This is the key result to develop a dimensional induction scheme for the asymptotic analysis of these operators near the singular set.
A complete description of Osserman four-manifolds whose Jacobi operators have a nonzero double root of the minimal polynomial is given.
The almost contact metric structure that we have on a real hypersurface $M$ in the complex quadric $Q^{m}=SO_{m+2}/SO_mSO_2$ allows us to define, for any nonnull real number $k$, the $k$-th generalized Tanaka-Webster connection on $M$, $\hat{\nabla}^{(k)}$. Associated to this connection we have Cho and torsion operators, $F_X^{(k)}$ and $T_X^{(k)}$, respectively, for any vector field $X$ tangent to $M$. From them and for any symmetric operator $B$ on $M$ we can consider two tensor fields of type (1,2) on $M$ that we will denote by $B_F^{(k)}$ and $B_T^{(k)}$, respectively. We will classify rea…
We prove that harmonic morphisms preserve the Jacobi operator along harmonic maps. We apply this result to prove infinitesimal and local rigidity (in the sense of Toth) of harmonic morphisms to a sphere.
A unified framework for photon and massive particle hypersurfaces in stationary spacetimes
本文提出了一种统一框架，将静态时空中的光子和巨粒子超曲面置于同一数学结构下，并探讨了相关Finsler结构的性质及洛伦兹力方程的解的存在性。
Let $f:M\ra \erre^{m+1}$ be an isometrically immersed hypersurface. In this paper, we exploit recent results due to the authors in \cite{bimari} to analyze the stability of the differential operator $L_r$ associated with the $r$-th Newton tensor of $f$. This appears in the Jacobi operator for the variational problem of minimizing the $r$-mean curvature $H_r$. Two natural applications are found. The first one ensures that, under the mild condition that the integral of $H_r$ over geodesic spheres grows sufficiently fast, the Gauss map meets each equator of $\esse^m$ infinitely many times. The se…
We study the Dirichlet problem at infinity on a Cartan-Hadamard manifold for a large class of operators containing in particular the p-Laplacian and the minimal graph operator.
Spectrum of the Laplacian and the Jacobi operator on Generalized rotational minimal hypersurfaces of spheres
本文研究球面广义旋转极小超曲面的谱，推导平均曲率方程，提出求拉普拉斯与稳定性算子特征值的方法，并以S^6中例子计算得到稳定性指标77、零化度14。
We prove a new upper bound for the first eigenvalue of the Dirac operator of a compact hypersurface in any Riemannian spin manifold carrying a non-trivial twistor spinor without zeros on the hypersurface. The upper bound is expressed as the first eigenvalue of a drifting Schrödinger operator on the hypersurface. Moreover, using a recent approach developed by O. Hijazi and S. Montiel, we completely characterize the equality case when the ambient manifold is the standard hyperbolic space.
We prove stability results associated with upper bounds for the first eigenvalue of certain second order differential operators of divergence-type on hypersurfaces of the Euclidean space. We deduce some applications to $r$-stability as well as to almost-Einstein hypersurfaces.
The eigenvalue problem for the Sen--Witten operator on closed spacelike hypersurfaces is investigated. The (square of its) eigenvalues are shown to be given exactly by the 3-surface integral appearing in the expression of the total energy-momentum of the matter+gravity systems in Witten's energy positivity proof. A sharp lower bound for the eigenvalues, given in terms of the constraint parts of the spacetime Einstein tensor, i.e. the energy and momentum densities of the matter fields, is given.
We introduce a new elliptic operator on null hypersurfaces of four-dimensional Lorentzian manifolds. This operator depends on the first and second fundamental forms of the sections of a foliation of the null hypersurface and its novelty originates from its covariant transformation under change of foliation. It thus provides at any point an elliptic structure intimately connected with the geometry of the null hypersurface, independent of the choice of a specific section through that point. No analytic or algebraic symmetries or other conditions are imposed on the metric. The spectral properties…
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