The Bochner Technique and Weighted Curvatures

On the Bochner Curvature Tensor in an Almost Hermitian Manifold

A classification theorem for RK-manifolds with linear dependence between invariants of an antiholomorphic plane in the tangent space is proved.

New Bochner type theorems

A classical theorem of Bochner asserts that the isometry group of a compact Riemannian manifold with negative Ricci curvature is finite. In this paper we give several extensions of Bochner's theorem by allowing "small" positive Ricci curvature.

On Bochner flat almost Kahler manifolds

The main purpose of this article is to prove that there exist no proper $AK_3$-manifold of dimension $2n\ge 6$ with vanishing Tricerri-Vanhecke Bochner curvature tensor and constant scalar curvature.

On some inequalities for submanifolds of Bochner Kaehler manifolds

B. Y. Chen established sharp inequalities between certain Riemannian invariants and the squared mean curvature for submanifolds in real space form as well as in complex space form. In this paper we generalize Chen inequalities for submanifolds of Bochner Kaehler manifolds. Moreover, we consider CR-warped product submanifolds of Bochner Kaehler manifold and establish an inequality for scalar curvature.

A Bochner Technique For Foliations With Non-Negative Transverse Ricci Curvature

We generalize the Bochner technique to foliations with non-negative transverse Ricci curvature. In particular, we obtain a new vanishing theorem for basic cohomology. Subsequently, we provide two natural applications, namely to degenerate 3-$(α,δ) $-Sasaki and certain Sasaki-$ η$-Einstein manifolds, which arise for example as Boothby-Wang bundles over hyperkähler and Calabi-Yau manifolds, respectively.

标量曲率的通用波纳切公式

A universal Bochner formula for scalar curvature

本文提出了一种适用于标量曲率的通用波纳切公式，涵盖了最小切片稳定性不等式、施罗德inger-利钦纳沃伊兹型公式及高维斯特恩水平集恒等式的特殊情况。

Ricci Curvature and Bochner Formulas for Martingales

We generalize the classical Bochner formula for the heat flow on M to martingales on the path space PM, and develop a formalism to compute evolution equations for martingales on path space. We see that our Bochner formula on PM is related to two sided bounds on Ricci curvature in much the same manner that the classical Bochner formula on M is related to lower bounds on Ricci curvature. Using this formalism, we obtain new characterizations of bounded Ricci curvature, new gradient estimates for martingales on path space, new Hessian estimates for martingales on path space, and streamlined proofs…

具有二次曲率项的 Weitzenböck-Bochner-Kodaira 公式

Weitzenböck-Bochner-Kodaira formulas with quadratic curvature terms

本文在紧 Kähler 流形上建立含二次曲率项的新型 Bochner-Kodaira 公式，并推导 Riemannian 与 Kähler 情形下的 Weitzenböck 公式，从而获得弱曲率条件下的消没定理与 Hodge 数估计。

Warped product Kaehler manifolds and Bochner-Kaehler metrics

Using as an underlying manifold an alpha-Sasakian manifold we introduce warped product Kaehler manifolds. We prove that if the underlying manifold is an alpha-Sasakian space form, then the corresponding Kaehler manifold is of quasi-constant holomorphic sectional curvatures with special distribution. Conversely, we prove that any Kaehler manifold of quasi-constant holomorphic sectional curvatures with special distribution locally has the structure of a warped product Kaehler manifold whose base is an alpha-Sasakian space form. Considering the scalar distribution generated by the scalar curvatur…

Bochner's technique for statistical structures

The main aim of this paper is to extend Bochner's technique to statistical structures. Other topics related to this technique are also introduced to the theory of statistical structures. It deals, in particular, with Hodge's theory, Bochner-Weitzenbock and Simon's type formulas. Moreover, a few global and local theorems on the geometry of statistical structures are proved, for instance, theorems saying that under some topological and geometrical conditions a statistical structure must be trivial. We also introduce a new concept of sectional curvature depending on statistical connections. On th…

曲率挤压条件下共形 Killing 向量场的 Bochner-Yano 型定理

Bochner-Yano Type Theorems for Conformal Killing Vector Fields under Curvature Pinching Conditions

研究曲率挤压条件下闭黎曼流形上共形 Killing 向量场，通过建立新 Bochner 型恒等式和 Moser 迭代得梯度估计，证明 Ricci 曲率有界时非平凡共形 Killing 向量场处处非零，推广了相关经典定理。

Bochner 技术的新曲率条件

New Curvature Conditions for the Bochner Technique

本文以曲率算子最低若干特征值平均值的下界为新条件，建立闭 Riemann 流形 Betti 数的消没与估计定理，推广经典结果并给出 3-正性例子。

The $\partial\overline{\partial}$-Bochner formulas for holomorphic mappings between Hermitian manifolds and their applications

In this paper, we derive some $\partial\overline{\partial}$-Bochner formulas for holomorphic maps between Hermitian manifolds. As applications, we prove some Schwarz lemma type estimates, rigidity and degeneracy theorems. For instance, we show that there is no non-constant holomorphic map from a comapct Hermitian manifold with positive (resp. non-negative) $\ell$-second Ricci curvature to a Hermitian manifold with non-positive (resp. negative) real bisectional curvature. These theorems generalize the results \cite{Ni1,Ni2} proved recently by L. Ni on Kähler manifolds to Hermitian manifolds. We…

Bochner-Weitzenböck formulas and curvature actions on Riemannian manifolds

Gradients are natural first order differential operators depending on Riemannian metrics. The principal symbols of them are related to the enveloping algebra and higher Casimir elements. We give certain relations in the enveloping algebra, which induce not only identities for higher Casimir elements but also all Bochner-Weitzenböck formulas for gradients. As applications, we give some vanishing theorems.

Bochner 技巧的一个 Thorpe 技巧

A Thorpe Trick for the Bochner Technique

本文证明，在维度至少为6时，具有适当正曲率算子的闭可定向黎曼流形必为实同调球，并特别处理了六维情形。

The Bochner formula for Riemannian flows

In this paper, we consider a Riemannian manifold (M, g) endowed with a Riemannian flow and we study the curvature term in the Bochner-Weitzenb{ö}ck formula of the basic Laplacian on M. We prove that this term splits into two parts. The first part depends mainly on the curvature operator of the underlying manifold M and the second part is expressed in terms of the O'Neill tensor of the flow. After getting a lower bound for this term, depending on these two parts, we establish an eigenvalue estimate of the basic Laplacian on basic forms. We then discuss the limiting case of the estimate and prov…

Local description of Bochner-flat (pseudo-)Kähler metrics

The Bochner tensor is the Kähler analogue of the conformal Weyl tensor. In this article, we derive local (i.e., in a neighbourhood of almost every point) normal forms for a (pseudo-)Kähler manifold with vanishing Bochner tensor. The description is pined down to a new class of symmetric spaces which we describe in terms of their curvature operators. We also give a local description of weakly Bochner-flat metrics defined by the property that the Bochner tensor has vanishing divergence. Our results are based on the local normal forms for c-projectively equivalent metrics. As a by-product, we also…

通过Calabi曲率算子刻画复射影空间

A characterization of complex projective space via the Calabi curvature operator

该研究针对Kähler-Einstein流形，利用含Calabi曲率算子的新Bochner公式证明了Tachibana型结果，分离出复射影空间，解决了AIM研讨会提出的相关问题。

On the Equivalence of the Entropic Curvature-Dimension Condition and Bochner's Inequality on Metric Measure Spaces

We prove the equivalence of the curvature-dimension bounds of Lott-Sturm-Villani (via entropy and optimal transport) and of Bakry--Émery (via energy and Γ_2$-calculus) in complete generality for infinitesimally Hilbertian metric measure spaces. In particular, we establish the full Bochner inequality on such metric measure spaces. Moreover, we deduce new contraction bounds for the heat flow on Riemannian manifolds and on mms in terms of the $L^2$-Wasserstein distance.

On the Bochner technique for singular distributions

In this paper we continue our recent study of a manifold endowed with a singular or regular distribution, determined as the image of the tangent bundle under a smooth endomorphism, and generalize Bochner's technique to the case of a distribution with a statistical type structure. Following the theory of statistical structures on Riemannian manifolds and construction of an almost Lie algebroid on a vector bundle, we define the modified statistical connection and exterior derivative on tensors. Then we introduce the Weitzenbock type curvature operator on tensors and derive the Bochner-Weitzenboc…

A Liouville-type theorem and Bochner formula for harmonic maps into metric spaces

We study analytic properties of harmonic maps from Riemannian polyhedra into CAT($κ$) spaces for $κ\in\{0,1\}$. Locally, on each top-dimensional face of the domain, this amounts to studying harmonic maps from smooth domains into CAT($κ$) spaces. We compute a target variation formula that captures the curvature bound in the target, and use it to prove an $L^p$ Liouville-type theorem for harmonic maps from admissible polyhedra into convex CAT($κ$) spaces. Another consequence we derive from the target variation formula is the Eells-Sampson Bochner formula for CAT(1) targets.

A Bochner Formula for Harmonic Maps into Non-Positively Curved Metric Spaces

We study harmonic maps from Riemannian manifolds into arbitrary non-positively curved and CAT(-1) metric spaces. First we discuss the domain variation formula with special emphasis on the error terms. Expanding higher order terms of this and other formulas in terms of curvature, we prove an analogue of the Eels-Sampson Bochner formula in this more general setting. In particular, we show that harmonic maps from spaces of non-negative Ricci curvature into non-positively curved spaces have subharmonic energy density. When the domain is compact the energy density is constant, and if the domain has…

Kohn-Rossi上同调与Bochner技巧

Kohn-Rossi cohomology and the Bochner technique

本文在具有非负曲率算子的紧致严格拟凸伪厄米特流形上证明了Betti数的消失定理，其核心方法是将Bochner技巧推广应用于CR流形。

A new proof of a classical result on the topology of orientable connected and compact surfaces by means of the Bochner technique

As an application of the Bochner formula, we prove that if a $2$-dimensional Riemannian manifold admits a non-trivial smooth tangent vector field $X$ then its Gauss curvature is the divergence of a tangent vector field, constructed from $X$, defined on the open subset out the zeroes of $X$. Thanks to the Whitney embedding theorem and a standard approximation procedure, as a consequence, we give a new proof of the following well-known fact: if on an orientable, connected and compact $2$-dimensional smooth manifold there exists a continuous tangent vector field with no zeroes, then the manifold …

Generalized Bochner formulas and Ricci lower bounds for sub-Riemannian manifolds of rank two

We study a new class of rank two sub-Riemannian manifolds encompassing Riemannian manifolds, CR manifolds with vanishing Webster-Tanaka torsion, orthonormal bundles over Riemannian manifolds, and graded nilpotent Lie groups of step two. These manifolds admit a canonical horizontal connection and a canonical sub-Laplacian. We construct on these manifolds an analogue of the Riemannian Ricci tensor and prove Bochner type formulas for the sub-Laplacian. As a consequence, it is possible to formulate on these spaces a sub-Riemannian analogue of the so-called curvature dimension inequality. Sub-Riema…

Integral Curvature Bounds and Betti Numbers

We introduce an upper bound of the Betti numbers of a compact Riemannian manifold given integral bounds on the average of the lowest eigenvalues of the curvature operator. We then establish a new curvature condition for the Betti numbers to vanish using the Bochner technique. This generalizes results from Gallot and Petersen-Wink.

Bochner-Weitzenböck formula and Li-Yau estimates on Finsler manifolds

We prove the Bochner-Weitzenböck formula for the (nonlinear) Laplacian on general Finsler manifolds and derive Li-Yau type gradient estimates as well as parabolic Harnack inequalities. Moreover, we deduce Bakry-Émery gradient estimates. All these estimates depend on lower bounds for the weighted flag Ricci tensor.

内蕴与正规平均里奇曲率：简单 d-向量的 Bochner--Weitzenboeck 恒等式

Intrinsic and Normal Mean Ricci Curvatures: A Bochner--Weitzenboeck Identity for Simple d-Vectors

本文定义内蕴与正规平均里奇曲率，推导简单 d-向量的 Bochner--Weitzenboeck 恒等式，并应用于调和向量消失准则与 Hodge Laplacian 特征值下界。

On Cosymplectic Conformal Connections

The aim of this paper is to introduce a cosymplectic analouge of conformal connection in a cosymplectic manifold and proved that if cosymplectic manifold M admits a cosymplectic conformal connection which is of zero curvature, then the Bochner curvature tensor of M vanishes.

Remarks on critical metrics of the scalar curvature and volume functionals on compact manifolds with boundary

We provide a general Böchner type formula which enables us to prove some rigidity results for $V$-static spaces. In particular, we show that an $n$-dimensional positive static triple with connected boundary and positive scalar curvature must be isometric to the standard hemisphere, provided that the metric has zero radial Weyl curvature and satisfies a suitable pinching condition. Moreover, we classify $V$-static spaces with non-negative sectional curvature.

Bochner-Kahler metrics

A Kahler metric is said to be Bochner-Kahler if its Bochner curvature vanishes. This is a nontrivial condition when the complex dimension of the underlying manifold is at least 2. In this article it will be shown that, in a certain well-defined sense, the space of Bochner-Kahler metrics in complex dimension n has real dimension n+1 and a recipe for an explicit formula for any Bochner-Kahler metric will be given. It is shown that any Bochner-Kahler metric in complex dimension n has local (real) cohomogeneity at most n. The Bochner-Kahler metrics that can be `analytically continued' to a complet…

半经典极限下 Bochner Laplacian 的预解式

Resolvents of Bochner Laplacians in the semiclassical limit

本文引入 Heisenberg 半经典伪微分算子，研究紧流形上线丛大幂截面空间，证明其包含 Bochner Laplacian，并在曲率非退化时得出其预解式及广义 Bergman 核等谱投影。

Some inequalities on Finsler manifolds with weighted Ricci curvature bounded below

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.

Lower Bound Estimates for The First Eigenvalue of The Weighted $p$-Laplacian on Smooth Metric Measure Spaces

New lower bounds of the first nonzero eigenvalue of the weighted $p$-Laplacian are established on compact smooth metric measure spaces with or without boundaries. Under the assumption of positive lower bound for the $m$-Bakry--Émery Ricci curvature, the Escober--Lichnerowicz--Reilly type estimates are proved; under the assumption of nonnegative $\infty$-Bakry--Émery Ricci curvature and the $m$-Bakry--Émery Ricci curvature bounded from below by a non-positive constant, the Li--Yau type lower bound estimates are given. The weighted $p$-Bochner formula and the weighted $p$-Reilly formula are deri…

从$\u200b\mathrm{RCD}(K,N)$空间到$\u200b\mathrm{CAT}(0)$空间的调和映射的Lipschitz连续性与Bochner-Eells-Sampson不等式

Lipschitz continuity and Bochner-Eells-Sampson inequality for harmonic maps from $\mathrm{RCD}(K,N)$ spaces to $\mathrm{CAT}(0)$ spaces

本文研究RCD(K,N)空间到CAT(0)空间调和映射的正则性，利用最优输运与热流相互作用及粘性摄动论证，证明其Lipschitz连续性并建立含Hessian项的Bochner-Eells-Sampson不等式，推广了经典理论。

Rigidity of automorphisms and spherical CR structures

We establish Bochner-type formulas for operators related to $CR$ automorphisms and spherical $CR$ structures. From such formulas, we draw conclusions about rigidity by making assumptions on the Tanaka-Webster curvature and torsion.

Anti-self-dual orbifolds with cyclic quotient singularities

An index theorem for the anti-self-dual deformation complex on anti-self-dual orbifolds with cyclic quotient singularities is proved. We present two applications of this theorem. The first is to compute the dimension of the deformation space of the Calderbank-Singer scalar-flat Kahler toric ALE spaces. A corollary of this is that, except for the Eguchi-Hanson metric, all of these spaces admit non-toric anti-self-dual deformations, thus yielding many new examples of anti-self-dual ALE spaces. For our second application, we compute the dimension of the deformation space of the canonical Bochner-…

子黎曼流形上的广义曲率维数不等式

Sub-Laplacian generalized curvature dimension inequalities on Riemannian foliations

本文在不假定度量束状结构和叶的总测地或最小条件的情况下，为一般黎曼流形的水平拉普拉斯算子发展了广义曲率维数不等式，并应用该框架得到多种几何和分析结果。

具有常Chern全纯截面曲率的紧致局部共形Kähler流形

Compact locally conformal Kähler manifolds with constant Chern holomorphic sectional curvature

本文证明了紧致局部共形Kähler流形上常全纯截面曲率猜想的Chern版本，通过曲率恒等式和Bochner-Kähler刚性，表明度量必为Kähler且为复空间形式。

An integral formula for affine connections

In this article, we introduce a $2$-parameter family of affine connections and derive the Ricci curvature. We first establish an integral Bochner technique. On one hand, this technique yields a new proof to our recent work in \cite{LX} for substatic manifolds. On the other hand, this technique leads to various geometric inequalities and eigenvalue estimates under a much more general Ricci curvature conditions. The new Ricci curvature condition interpolates between static Ricci tensor and $1$-Bakry-Emery Ricci, and also includes the conformal Ricci as an intermediate case.

曲率为零条件下黎曼挠率空间的刚性结果

Rigidity results for Riemannian twistor spaces under vanishing curvature conditions

本文利用活动标架法研究四维黎曼流形挠率空间的刚性，证明$\mathbb{CP}^3$是唯一Bochner张量平行者，分类相关空间并证明共形平坦情形不存在，同时推广Atiyah–Hitchin–Singer自对偶结果。

Rigidity theorems of complete Kähler-Einstein manifolds and complex space forms

We derive some elliptic differential inequalities from the Weitzenböck formulas for the traceless Ricci tensor of a Kähler manifold with constant scalar curvature and the Bochner tensor of a Kähler-Einstein manifold respectively. Using elliptic estimates and maximum principle, some $L^p$ and $L^\infty $ pinching results are established to characterize Kähler-Einstein manifolds among Kähler manifolds with constant scalar curvature, and others are given to characterize complex space forms among Kähler-Einstein manifolds. Finally, these pinching results may be combined to characterize complex spa…

在下界1加权里奇曲率限制下加权极小超曲面的稳定性

Stability of weighted minimal hypersurfaces under a lower $1$-weighted Ricci curvature bound

本文研究了在下界1加权里奇曲率限制下加权极小超曲面的稳定性，推导了相关几何结论，并得出三维加权流形的结构定理及非存在性结果。

Weighted Projective Ricci Curvature in Finsler Geometry

In this paper, we introduce the weighted projective Ricci curvature as an extension of projective Ricci curvature introduced by Z. Shen. We characterize the class of Randers metrics of weighted projective Ricci flat curvature. We find the necessary and sufficient condition under which a Kropina metric has weighted projective Ricci flat curvature. Finally, we show that every projectively flat metric with isotropic weighted projective Ricci and isotropic S-curvature is a Kropina metric or Randers metric.

Comparison geometry of manifolds with boundary under a lower weighted Ricci curvature bound

We study Riemannian manifolds with boundary under a lower weighted Ricci curvature bound. We consider a curvature condition in which the weighted Ricci curvature is bounded from below by the density function. Under the curvature condition, and a suitable condition for the weighted mean curvature for the boundary, we obtain various comparison geometric results.

低0-加权Ricci曲率下界下加权流形的几何分析

Geometric analysis on weighted manifolds under lower $0$-weighted Ricci curvature bounds

本研究在低0-加权Ricci曲率下界下开展加权黎曼流形几何分析，证明了两类流形上的特征值估计，还推导出ABP估计与Brendle型Sobolev不等式。

具有正加权旗曲率的Finsler流形

Finsler manifolds with Positive Weighted Flag Curvature

该研究引入基于$T$-曲率修正的加权旗曲率概念，证明两类具正/非负加权旗曲率的Finsler流形分别微分同胚于欧氏空间与欧氏球，推进了Finsler几何的曲率与拓扑关联研究。

On the first stability eigenvalue of surfaces with constant weighted mean curvature

Let $Σ$ be a compact immersed surface with constant weighted mean curvature $H_f$ in a weighted manifold $(M^3,g,f)$. In this paper we obtain upper bounds for the first eigenvalue of the weighted Jacobi operator on $Σ$ in terms of $H_f$ and the curvature of the ambient. As consequence we obtain that there is no stable self-shrinker of the mean curvature flow.

Positive weighted sectional curvature

In this paper, we give a new generalization of positive sectional curvature called positive weighted sectional curvature. It depends on a choice of Riemannian metric and a smooth vector field. We give several simple examples of Riemannian metrics which do not have positive sectional curvature but support a vector field that gives them positive weighted curvature. On the other hand, we generalize a number of the foundational results for compact manifolds with positive sectional curvature to positive weighted curvature. In particular, we prove generalizations of Weinstein's theorem, O'Neill's fo…

The weighted connection and sectional curvature for manifolds with density

In this paper we study sectional curvature bounds for Riemannian manifolds with density from the perspective of a weighted torsion free connection introduced recently by the last two authors. We develop two new tools for studying weighted sectional curvature bounds: a new weighted Rauch comparison theorem and a modified notion of convexity for distance functions. As applications we prove generalizations of theorems of Preissman and Byers for negative curvature, the (homeomorphic) quarter-pinched sphere theorem, and Cheeger's finiteness theorem. We also improve results of the first two authors …

The weighted mixed curvature of a foliated manifold

In this paper, we introduce the weighted mixed (sectional, Ricci and scalar) curvature of a foliated (and almost-product) Riemannian manifold $(M,g)$ equipped with a vector field $X$. We define several functions ($q$th Ricci type curvatures), which "interpolate" between the weighed sectional and Ricci curvatures. The novel concepts of the "mixed curvature-dimension" condition and "synthetic dimension of a distribution" allow us to update the estimate of the diameter of a compact Riemannian foliation and to prove new splitting theorems for almost-product manifolds of nonnegative/nonpositive wei…

加权Sasaki球面的新曲率刻画

New curvature characterization for the weighted Sasaki sphere

本文利用Yang和Zhang的移位锥分类Sasaki流形，从而给出加权Sasaki球面的新曲率刻画。

Volume estimates and classification theorem for constant weighted mean curvature hypersurfaces

In this paper, we prove a classification for complete embedded constant weighted mean curvature hypersurfaces $Σ\subset\mathbb{R}^{n+1}$. We characterize the hyperplanes and generalized round cylinders by using an intrinsic property on the norm of the second fundamental form. Furthermore, we prove an equivalence of properness, finite weighted volume and exponential volume growth for submanifolds with weighted mean curvature of at most linear growth.

Constant weighted mean curvature hypersurfaces in Shrinking Ricci Solitons

In this paper, we study constant weighted mean curvature hypersurfaces in shrinking Ricci solitons. First, we show that a constant weighted mean curvature hypersurface with finite weighted volume cannot lie in a region determined by a special level set of the potential function, unless it is the level set. Next, we show that a compact constant weighted mean curvature hypersurface with a certain upper bound or lower bound on the mean curvature is a level set of the potential function. We can apply both results to the cylinder shrinking Ricci soliton ambient space. Finally, we show that a consta…