The Lichnerowicz theorem on CR manifolds
Elisabetta Barletta
得到了紧严格伪凸CR流形的次拉普拉斯算子谱的Bochner型公式和下式估计。
We obtain a Bochner type formula and an estimate from below on the spectrum of the sublaplacian of a compact strictly pseudoconvex CR manifold.
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An Introduction to Harmonic Manifolds and the Lichnerowicz Conjecture
标题不言自明。我们的目标是提供一个易于阅读和自我包含的介绍领域的调和流形。只需要基本的黎曼几何知识。在给出了调和性的定义和一些性质之后，我们集中讨论了Z。I. Lichnerowicz猜想在紧致单连通流形类中的Szabó证明
A Lichnerowicz-type result on a seven-dimensional quaternionic contact manifold
本文建立了一个类似于经典Lichnerowicz定理的定理，给出了紧致七维四元数切触流形上次Laplacian的第一非零特征值的一个精确下界，并假定了qc-Ricci张量、挠率张量及其特殊的共变导数的一个下界.
Given a compact, three-dimensional, real-analytic Lorentzian manifold $(M,g)$, we prove that the identity component of the conformal group preserves a metric in the conformal class $[g]$, or $(M,g)$ is conformally flat.
A Lichnerowicz vanishing theorem for proper cocompact actions
当作用群为幺模时，我们建立了具有真余紧作用的非紧自旋流形的Lichnerowicz型消灭定理。
We show a Lichnerowicz-Obata type estimate for the first eigenvalue of the Laplacian of $n$-dimensional closed Riemannian manifolds with an almost parallel $p$-form ($2\leq p \leq n/2$) in $L^2$-sense, and give an almost decomposition result of the manifold under some pinching conditions when $2\leq p<n/2$.
The Lichnerowicz-Obata theorem on sub-Riemannian manifolds with transverse symmetries
证明了具有横对称的次黎曼流形上次拉普拉斯算子第一特征值的一个下界。当流形为H型时，我们得到了相应的刚性结果：若第一特征值的最优下界达到，则流形等价于1或3-Sasakian球面.
A note on the Lichnerowicz vanishing theorem for proper actions
本文证明了一个Lichnerowicz型消灭定理，它适用于具有真余紧作用的非紧自旋流形。这推广了刘自然的一个结果，他证明了作用群是幺模的情形。
In this paper, we show several rigidity results for harmonic $(p,q)$-forms in complete Kähler manifolds. We also give several applications to study non-compact Kähler manifolds with parallel Bochner tensor or quaternion Kähler manifolds. Our results are natural extensions of Petersen and Wink's results in \cite{PW21, PW} in the setting of complete, non-compact Kähler manifolds.
On a Lichnerowicz type cohomology attached to a function
本文定义了光滑流形上一个新的上同调，称为Lichnerowicz型上同调。首先，我们研究了这种上同调的一些基本性质：de Rham型同构，对函数的依赖性，奇异形式，相对上同调，Mayer-Vietoris序列，同伦不变性，其次，考虑了正则情形。这些概念是使用技术从研究两个上同调的光滑流形：Lichnerowicz上同调和上同调附加到一个功能。
Lichnerowicz-Jacobi cohomology and homology of Jacobi manifolds: modular class and duality
讨论了Jacobi流形的Lichnerowicz-Jacobi上同调和同调。我们目前都在一个统一的方法使用的代表性的李代数的功能本身的哈密尔顿向量场。使用相关的李代数体可以证明Lichnerowicz-Jacobi上同调和同调在Jacobi结构的共形变化下是不变的，并在模类消失时建立Lichnerowicz-Jacobi上同调和同调之间的对偶。我们还计算了Lichnerowicz-Jacobi上同调和同调的各种各样的例子。
Local normal forms for c-projectively equivalent metrics and proof of the Yano-Obata conjecture in arbitrary signature. Proof of the projective Lichnerowicz conjecture for Lorentzian metrics
复流形上的两个Kähler度量如果它们的$J$ -平面曲线重合，则称为c-投射等价.这些曲线由加速度与速度成复比例的性质定义。本文给出了任意签名的所有c-投射等价Kähler度量对的一个显式局部描述，并利用这个描述证明了经典的Yano-Obata猜想：证明了在任意签名的闭连通Kähler流形上，任何c-投射向量场都是仿射向量场，除非该流形是具有Fubini-Study度量（的倍数）的$CP^n$.作为一个副产品，我们证明了投影Lichnerowicz猜想的度量的洛伦兹签名：我们表明，在一个封闭的连通洛伦兹流形上，任何投影向量场是一个仿射向量场。
A Lichnerowicz estimate for the first eigenvalue of convex domains in Kähler manifolds
本文证明了Kähler流形的紧致凸区域的Lichnerowicz估计，其中里奇曲率满足$\Ric \ge k$，对某个常数$k>0$ .当达到相等时，域的边界是全测地线，并且存在非平凡的全纯向量场。我们证明了在复射影空间中一个半径足够大的球提供了一个强伪凸域的例子，它不是凸的，并且Lichnerowicz估计失败。
We construct examples of nondegenerate CR manifolds with Levi form of signature $(p,q)$, $2\leq p\leq q$, which are compact, not locally CR flat, and admit essential CR vector fields. We also construct an example of a noncompact nondegenerate CR manifold with signature $(1,n-1)$ which is not locally CR flat and admits an essential CR vector fields. These provide counterexamples to the analogue of the Lichnerowicz conjecture for CR manifolds with mixed signature.
Lichnerowicz-type equations on complete manifolds
在适当的谱假设下，证明了完备流形上Lichnerowicz型方程正解的两个存在性结果。我们还给出了先验界和比较结果，立即产生某些类的解决方案的唯一性。在我们的分析中不涉及曲率假设。
Relationship between sectional curvature and null spaces of Lichnerowicz-type Laplacians and their smallest eigenvalues
该研究修正了首版错误，证明截面曲率严格正且Ricci曲率适当夹逼的黎曼流形第二类曲率算子严格正，还得到三类拉普拉斯算子零空间的消没定理及最小特征值估计。
Lichnerowicz-type equations with sign-changing nonlinearities on complete manifolds with boundary
证明了完备流形上具有边界和非线性Neumann条件的Lichnerowicz型方程正解的存在性定理。这类非线性问题是在共形方法框架下研究广义相对论Einstein-标量场方程解时很自然地出现的。
The Lichnerowicz and Obata first eigenvalue theorems and the Obata uniqueness result in the Yamabe problem on CR and quaternionic contact manifolds
本文介绍了亚黎曼CR几何和四元数接触几何中某些问题的研究进展。重点是圆球面上相应的Yamabe问题，Lichnerowicz-Obata第一特征值估计，以及这两个问题之间的关系。一个动机从黎曼的情况下，突出了新的和旧的想法，然后开发的岩泽分黎曼几何的设置。
There are very few general theorems on the kernel of the well-known Lichnerowicz Laplacian. In the present article we consider the geometry of the kernel of this operator restricted to covariant (not necessarily symmetric or skew-symmetric) tensors. Our approach is based on the analytical method, due to Bochner, of proving vanishing theorems for the null space of Laplace operator. In particular, we pay special attention to the kernel of the Lichnerowicz Laplacian on Riemannian symmetric spaces of compact and noncompact types. In conclusion, we give some applications to the theories of infinite…
Heat flow method to Lichnerowicz type equation on closed manifolds
本文在闭流形中建立了如下类型的Lichnerowicz方程-Δu=A（x）u^{-p}-B（x）u^{q}，\quad在\quad M中正解的存在性结果，其中$p>1, q>0$，$A(x)>0$，$B(x)\geq0$是光滑函数.我们的分析是基于以下热方程{ll} u_t-Δu=A（x）u^{-p}-B（x）u^{q}，\quad在\quad M中\times \mathbb{R}^{+}，u（x，0）=u_0，\quad在\quad M中，具有正光滑初始值$u_0$的整体正解的存在性.
A variational analysis of Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds
建立了紧流形上Einstein-标量场Lichnerowicz方程正解的存在性和不存在性的新结果。该方程源于广义相对论中爱因斯坦-标量场系统的哈密顿约束方程。我们的分析介绍变分技术，在山路引理的形式，分析的哈密顿约束方程，这是以前研究过的其他方法。
We prove rigidity for the Lichnerowicz-type eigenvalue estimate for the Kohn Laplacian on strictly pseudoconvex three-manifolds with nonnegative CR Paneitz operator and positive Webster curvature.
For a second order operator on a compact manifold satisfying the strong Hörmander condition, we give a bound for the spectral gap analogous to the Lichnerowicz estimate for the Laplacian of a Riemannian manifold. We consider a wide class of such operators which includes horizontal lifts of the Laplacian on Riemannian submersions with minimal leaves.
We study the prescribed scalar curvature problem, namely finding which function can be obtained as the scalar curvature of a metric in a given conformal class. We deal with the case of asymptotically hyperbolic manifolds and restrict ourselves to non positive prescribed scalar curvature. Following earlier results, we obtain a necessary and sufficient condition on the zero set of the prescribed scalar curvature so that the problem admits a (unique) solution.
Roe C*-algebra for groupoids and generalized Lichnerowicz Vanishing theorem for foliated manifolds
对于单位空间一般不紧的局部紧广群，引入Roe C* 代数-代数的概念，并赋予它适当的粗结构和Haar系.利用Connes的切群胚方法，在具有额外度量结构的李群胚上引入椭圆微分算子的解析指标，该指标取Roe C* 代数的K-理论中的值.我们应用我们的理论导出了开流形上的叶理的一个Lichnerowicz型消失结果。
A Lichnerowicz equation in the Einstein-scalar field theory on non-CMC closed manifolds
本文针对非常平均曲率闭流形上爱因斯坦-标量场论中的利希诺维茨方程，证明其存在正且本质有界的解，采用含次超解的不动点论证，并给出阻止正经典解存在的系数条件。
Hodge-de Rham and Lichnérowicz Laplacians on double forms and some vanishing theorems
该研究在双形式框架下证明Lichnérowicz拉普拉斯是两类Hodge-de Rham推广拉普拉斯的平均，引入新乘积建立曲率项无指标公式，得到多类消没定理，推广Petersen-Wink结果并揭示曲率算子特征秩对流形结构的影响。
We prove a Lichnerowicz type lower bound for the first nontrivial eigenvalue of the $p$-Laplacian on Kähler manifolds. Parallel to the $p = 2$ case, the first eigenvalue lower bound is improved by using a decomposition of the Hessian on Kähler manifolds with positive Ricci curvature.
In this paper, we give two Lichnerowicz type formulas for modified Novikov operators. We prove KastlerKalau-Walze type theorems for modified Novikov operators on compact manifolds with (resp.without) boundary. We also compute the spectral action for Witten deformation on 4-dimensional compact manifolds.
Ricci curvature and eigenvalue estimates for the magnetic Laplacian on manifolds
本文给出了具有磁势的闭黎曼流形上磁拉普拉斯算子的Lichnerowicz型估计和（高阶）Buser型估计.这些结果涉及本征值，磁场，里奇曲率，和Cheeger型常数。
In this paper, we obtain two Lichnerowicz type formulas for sub-signature operators. And we give the proof of Kastler-Kalau-Walze type theorems for sub-signature operators on 4-dimensional and 6-dimensional compact manifolds with (resp.without) boundary.
In this paper, we obtain two Lichnerowicz type formulas for the Dirac-Witten operators. And we give the proof of Kastler-Kalau-Walze type theorems for the Dirac-Witten operators on 4-dimensional and 6- dimensional compact manifolds with (resp.without) boundary
Supersymmetry and the generalized Lichnerowicz formula
Lichnerowicz [8]的一个经典结果涉及黎曼流形$M$上Clifford模${\cal E}$ \上Clifford联络定义的Dirac算子的平方的分解.最近，这个公式被推广到任意Dirac算子[2]。在本文中，我们证明了一个超对称版本的广义Lichnerowicz公式，动机的事实，有一个一对一的对应关系Clifford超连接和Dirac算子。我们推广这个结果，得到一个简单的公式的超曲率的广义Bismut超连接。这可以被看作是证明任意狄拉克算子族的局部指数定理的第一步。
In this paper, we give the Lichnerowicz type formulas for statistical de Rham Hodge operators. Moreover, Kastler-Kalau-Walze type theorems for statistical de Rham Hodge operators on compact manifolds with (respectively without) boundary are proved.
In Kähler geometry, Fujiki--Donaldson show that the scalar curvature arises as the moment map for Hamiltonian diffeomorphisms. In generalized Kähler geometry, one does not have suitable notions of Levi-Civita connection and curvature, however there still exists a precise framework for a moment map and the scalar curvature is defined as the moment map. Then a fundamental question is to understand the existence or non-existence of generalized Kähler structures with constant scalar curvature. In the paper, we study the Lie algebra of automorphisms of a generalized complex manifold. We assume that…
A Lichnerowicz Vanishing Theorem for the Maximal Roe Algebra
本文证明了在适当条件下，Dirac算子的极大高阶指标在极大等变Roe代数的K理论中消没，其关键是在极大等变一致Roe代数中建立泛函演算。
We establish some important inequalities under the condition that the weighted Ricci curvature $\mathrm{Ric}_{\infty}\geq K$ for some constant $K >0$ by using improved Bochner inequality and its integrated form. Firstly, we obtain a sharp Poincaré-Lichnerowicz inequality. Further, we give a new proof for logarithmic Sobolev inequality. Finally, we obtain an estimate of the volume of geodesic balls.
Structure theorems for Lichnerowicz-sharp graphs
该研究针对离散比较几何中的Lichnerowicz-尖锐图，证明移除特定边后所得图的规范丛结构，推导顶点度恒定时为超立方体丛的结论，并构造了非超立方体纤维的此类图，丰富了正曲率离散图的结构理论。
Let g be a G-invariant Einstein metric on a compact homogeneous space M=G/K. We use a formula for the Lichnerowicz Laplacian of g at G-invariant TT-tensors to study the stability type of g as a critical point of the scalar curvature function. The case when g is naturally reductive is studied in special detail.
In this paper, we establish an intrinsic Gauss-Bonnet-Chern formula for Finsler manifolds by using the Mathai-Quillen's superconnection formalism, in which no extra vector field is involved. Furthermore, we prove a more general Lichnerowicz formula in this direction through a geometric localization procedure.
Gradient estimates for some $f$-heat equations driven by Lichnerowicz's equation on complete smooth metric measure spaces
给定完备光滑度量测度空间$(M,g,e^{-f}dv)$，且Bakry-Émery里奇曲率有界，研究了以下一般$f$ -热方程$$ u_t=Δ_f u+au\log u+bu +Au^p+Bu^{-q} $$和\[ u_t=Δ_f u+Ae^{pu}+Be^{-pu}+D \]解的各种梯度估计.作为副产品，我们得到了几个非线性方程正解的Liouville型定理和Harnack型不等式，其中包括薛定谔方程，Yamabe方程和Lichnerowicz型方程作为特例.
Examples of non $d_ω$-exact locally conformal symplectic forms
本文给出了文献[1]中所考虑的解流形$M_{n,k}$上的两个三参数局部共形辛形式族，并利用Lichnerowicz上同调的Hodge-de Rham理论证明了它们不是$d_ω$精确的，即它们的Lichnerowicz类是非平凡的（定理1）.这有几个重要的几何后果（推论2，3）。这也意味着相应的局部共形辛结构的自同构自同构群的行为很像紧辛流形的辛同构群。我们开始分类的局部共形辛形式在每个3-参数家庭（定理2，推论1）。我们还证明了第一（和）第三Lichnerowicz上同调类是非零的（定理3）。最后，我们观察到的流形$M_{n,k}$进行几个有趣的叶理和泊松结构。
For any $G$-invariant metric on a compact homogeneous space $M=G/K$, we give a formula for the Lichnerowicz Laplacian restricted to the space of all $G$-invariant symmetric $2$-tensors in terms of the structural constants of $G/K$. As an application, we compute the $G$-invariant spectrum of the Lichnerowicz Laplacian for all the Einstein metrics on most generalized Wallach spaces and any flag manifold with $b_2(M)=1$. This allows to deduce the $G$-stability and critical point types of each of such Einstein metrics as a critical point of the scalar curvature functional.
Eigenvalue estimates and L1 energy on closed manifolds
本文研究了带权测度的闭流形上漂移Laplacian算子特征值的Lichnerowicz型估计和漂移热方程的L1和L2能量.从某种意义上说，本文的研究是关于Ricci孤子的本征值估计。
Vanshing Theorems for Quaternionic Kaehler Manifolds
讨论了黎曼流形的和乐群的表示论、形式上的Hodge-Laplace算子的Weitzenboeck公式和扭曲Dirac算子的Lichnerowicz公式之间的特殊相互作用.对于四元数Kaehler流形，这导致狄拉克和拉普拉斯算子的特征值估计的简单证明。此外，它使我们能够确定哪些表示可以有助于谐波形式。作为推论，我们证明了在具有负数量曲率的紧致四元数Kaehler流形上某些奇贝蒂数为零.我们简化了几个相关结果的证明。
Noncompact harmonic manifolds
Lichnerowicz猜想断言所有调和流形要么是平坦的，要么是秩为1的局部对称空间。这个猜想已由Z. I.具有紧泛覆盖的调和流形的Szabo。E. Damek和F.利玛窦提供的例子表明，在非紧的情况下，猜想是错误的。然而，这样的流形不允许紧商。所有非紧调和空间的分类仍然是一个非常困难的公开问题。本文综述了非紧单连通调和流形的最新结果，并证明了许多新的结果，包括一般的非紧调和流形和具有纯指数体积增长的非紧调和流形.
We establish some important inequalities under a lower weighted Ricci curvature bound on Finsler manifolds. Firstly, we establish a relative volume comparison of Bishop-Gromov type. As one of the applications, we obtain an upper bound for volumes of the Finsler manifolds. Further, when the S-curvature is bounded on the whole manifold, we obtain a theorem of Bonnet-Myers type on Finsler manifolds. Finally, we obtain a sharp Poincaré-Lichnerowicz inequality by using integrated Bochner inequality, from which we obtain a sharp lower bound for the first eigenvalue on the Finsler manifolds.
In this paper, we give a Lichnerowicz type formula for the J-twist D_J of the Dirac operator. And we prove a Kastler-Kalau-Walze type theorem for the J-twist D_J of the Dirac operator on 3-dimensional and 4-dimensional almost product Riemannian spin manifold with boundary.
In this paper, we give Lichnerowicz type formulas for the perturbation of the de Rham Hodge operator. We prove the Kastler-Kalau-Walze type theorems for the perturbation of the de Rham Hodge operator on 4-dimensional and 6-dimensional compact manifolds with (resp.without) boundary. Some concrete examples of the perturbation of the de Rham Hodge operator are provided for our main theorems.
Finite-energy solutions to Einstein-scalar field Lichnerowicz equations on complete Riemannian manifolds
本文研究了完备黎曼流形上爱因斯坦-标量场利钦诺维奇方程的有限能量解的存在性，通过正则化问题、山峰点论证和Harnack不等式，证明了在特定条件下非负解和正解的存在性。
The Kastler-Kalau-Walze type theorems about J-Witten deformation
本文针对J-Witten变形建立Lichnerowicz型公式，并在四维与六维近乘积黎曼旋量流形上证明了带边界和不带边界情形下的Kastler-Kalau-Walze型定理。
In this paper, we obtain two Lichnerowicz type formulas for the operators $\sqrt{-1}\widehat{c}(V)(d+δ)$ and $-\sqrt{-1}(d+δ)\widehat{c}(V)$. And we give the proof of Kastler-Kalau-Walze type theorems for the operators $\sqrt{-1}\widehat{c}(V)(d+δ)$ and $-\sqrt{-1}(d+δ)\widehat{c}(V)$ on 3,4-dimensional oriented compact manifolds with (resp.without) boundary.
Splitting theorems for Finsler manifolds of nonnegative Ricci curvature
研究了包含直线的非负加权里奇曲率的Finsler流形的结构，推广了经典的Cheeger-Gromoll-Lichnerowicz分裂定理.这样一个空间一般容许一个保纯的、保测度的分裂。对于一类特殊的Berwald空间，当Busemann函数的梯度向量场生成一个单参数等距族时，我们可以进行等距分裂.对Berwald空间也给出了一个贝蒂数估计.
Adiabatic Limits and Foliations
我们利用绝热极限来研究叶状流形。Bott联系自然地表现为Levi-Civita联系的绝热极限。作为应用，我们构造了一些与叶状有关的自然椭圆算子，并给出了Connes[Co]的消失定理的一个直接几何证明，该定理将Lichnerowicz消灭定理[L]推广到了具有自旋叶的叶状流形上，我们称之为几乎黎曼叶状.利用我们的方法还证明了几个新的消灭定理。
A Kastler-Kalau-Walze type theorem for the $J$-twist of the Dirac operator with torsion
该研究推导了带挠率的Dirac算子的$J$-扭的Lichnerowicz型公式，并在4维和6维带边界近乘积黎曼自旋流形上证明了对应的Kastler-Kalau-Walze型定理，拓展了非交换几何相关结果。
A pinching estimate for solutions of the linearized Ricci flow system on 3-manifolds
证明了闭三维流形上线性化里奇流系统解的一个估计。这个估计是Hamilton的pinching是3-流形上具有正里奇曲率的里奇流解的Ricci曲率的保持估计的推广。在我们的估计中，我们对初始度量的曲率没有任何假设。我们表明，规范的Lichnerowicz拉普拉斯热方程（耦合到里奇流）的解决方案是有界的一个常数（取决于时间）乘以数量曲率加上一个常数。这依赖于Bochner型公式，并建立6个变量的4次齐次多项式的非负性。
Eigenvalue estimates for a class of elliptic differential operators on compact manifolds
本文的动机是研究一个自然出现在黎曼几何中的二阶椭圆算子，例如在研究具有常数$r$ -中曲率的超曲面中。我们证明了这类算子的一个推广的Bochner型公式，作为应用，我们得到了两种特殊情况下第一非零特征值的精确估计。这些结果可以看作是Lichnerowicz-Obata定理的推广。
We show that a compact quaternionic contact manifold of dimension seven that satisfies a Lichnerowicz-type lower Ricci-type bound and has the $P$-function of any eigenfunction of the sub-Laplacian non-negative achieves its smallest possible eigenvalue only if the structure is qc-Einstein. In particular, under the stated conditions, the lowest eigenvalue is achieved if and only if the manifold is qc-equivalent to the standard $3$-Sasakian sphere.
Positive scalar curvature and the Euler class
我们证明了经典Lichnerowicz消灭定理的推广：如果$F$是闭自旋流形$M$上的定向平坦向量丛，使得$TM$带有正的数量曲率度量，则有$<\widehat A(TM)e(F),[M]>=0$，其中$e(F)$是$F$的Euler类.
Twisted Dirac Operators and the Noncommutative Residue for Manifolds with Boundary
本文给出了Dirac算子和签名算子被具有非幺正联络的向量丛扭曲的两个Lichnerowicz型公式。我们还证明了四维流形上的扭Dirac算子和扭签名算子的两个Kastler-Kalau-Walze型定理，其中（分别）没有边界。
The Einstein-scalar field constraints on asymptotically Euclidean manifolds
我们利用保角方法得到了Einstein-标量场引力约束方程的解。处理标量场比处理物质场（如流体、麦克斯韦场或杨-米尔斯场）更具挑战性，因为标量场在Lichnerowicz方程中引入了三个额外的项，而不是一个。我们的证明是建设性的，并允许任意维数（>2）以及低正则性的初始数据。
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