Weimin Sheng, Lisheng Wang
In this paper, we investigate a class of quadratic Riemannian curvature functionals on closed smooth manifold $M$ of dimension $n\ge 3$ on the space of Riemannian metrics on $M$ with unit volume. We study the stability of these functionals at the metric with constant sectional curvature as its critical point.
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The objet of this paper is the study of variations of a functional whose integrant is the $σ_u$-curvature of closed submanifolds of arbitrary codimension in Riemannian manifolds.
Four-dimensional homogeneous critical metrics for quadratic curvature functionals
本文确定了四维空间中所有使二次曲率泛函达到临界的齐次度量。
Manifolds with quadratic curvature decay and fast volume growth
给出了球面空间型上曲率二次衰减的非紧黎曼流形具有端点为锥的有限拓扑型的充分条件.
On manifolds with quadratic curvature decay
给出了具有二次曲率衰减的完备非紧流形具有有限拓扑型的条件。特别是，我们发现之间的联系的拓扑结构的流形与二次曲率衰减和一些性质的渐近锥面这样的流形。
Motivated by recent progress on a spinorial analogue of the Yamabe problem in the geometric literature, we study a conformally invariant spinor field equation on the $m$-sphere, $m\geq2$. Via variational methods and the spinorial Weierstraß representation, we study the problem of prescribing mean curvature for the immersion $S^2\to\mathbb{R}^3$.
We study rigidity results for the Einstein metrics as the critical points of a family of known quadratic curvature functionals involving the scalar curvature, the Ricci curvature and the Riemannian curvature tensor, characterized by some pointwise inequalities involving the Weyl curvature and the traceless Ricci curvature. Moreover, we also provide a few rigidity results for locally conformally flat critical metrics.
We study closed $n$-dimensional manifolds of which the metrics are critical for quadratic curvature functionals involving the Ricci curvature, the scalar curvature and the Riemannian curvature tensor on the space of Riemannian metrics with unit volume. Under some additional integral conditions, we classify such manifolds. Moreover, under some curvature conditions, the result that a critical metric must be Einstein is proved.
Variational properties of the total inverse mean curvature in the plane under boundary constraints
该研究针对半平面内带指定边界的曲线，分析其总逆平均曲率的变分行为，明确了给定面积下临界点的存在性与强稳定性，并据此推导出局部极小性性质，丰富了平面曲线变分问题的相关结论。
The objet of this paper is the study of the variations of a functional whose integrant is the r-th weighted curvature on the hypersurface of a closed Riemannian manifold. Some applications to hypersurfaces of the Euclidean space and the unit round sphère are given.
An example of compact Kähler manifold with nonnegative quadratic bisectional curvature
我们构造了一个非负二次对分曲率的紧致Kähler流形，它不允许任何非负正交对分曲率的Kähler度量.该流形是一个第二贝蒂数等于1的7维Kähler C-空间，其正则度量是一个正数量曲率的Kähler-Einstein度量
Manifolds with Quadratic Curvature Decay and Slow Volume Growth
我们证明了对于非紧流形，存在拓扑障碍，使得它可以接受一个曲率二次衰减的黎曼度量和一个比同维数的欧几里得空间慢的体积增长.
Sobolev Inequality In Manifolds With Lower Quadratic Curvature Decay
本文利用ABP方法，在具有下二次曲率衰减的完备黎曼流形中，为紧致区域和子流形建立了Sobolev不等式。
In this paper, we consider some rigidity results for the Einstein metrics as the critical points of some known quadratic curvature functionals on complete manifolds, characterized by some point-wise inequalities. Moreover, we also provide rigidity results by the integral inequalities involving the Weyl curvature, the trace-less Ricci curvature and the Sobolev constant, accordingly.
In this paper, we study Riemannian functionals defined by $L^2$-norms of Ricci curvature, scalar curvature, Weyl curvature, and Riemannian curvature. We try to understand stability of their critical points that are products of Einstein metrics. In particular, we prove that the product of a spherical space form and a compact hyperbolic manifold is unstable for some quadratic functionals if the first eigenvalue of the Laplacian of the hyperbolic manifold is sufficiently small.
In this paper, we prove some rigidity results for the Einstein metrics as the critical points of a family of known quadratic curvature functionals on closed manifolds, characterized by some point-wise inequalities. Moreover, we also provide a few rigidity results that involve the Weyl curvature, the trace-less Ricci curvature and the Yamabe invariant, accordingly.
Ricci solitons as critical points of quadratic curvature functionals
研究Ricci孤子作为特殊二次曲率泛函临界点的刚性，分析其与Einstein度量的关系。
In this paper, we construct a complete n-dim Riemannian manifold with positive Ricci curvature, quadratically nonnegatively curved infinity and infinite topological type. This gives a negative answer to a conjecture by Jiping Sha and Zhongmin Shen in the case of n greater than or equal to 6.
Variational formulas of higher order mean curvatures
本文建立了$p=0,1,...,[\frac{n}{2}]$的一般黎曼流形$N^{n+m}$中子流形$M^n$的全$2p$阶平均曲率泛函$\mathcal {M}_{2p}$的第一变分公式及其Euler-Lagrange方程.作为例子，我们证明了复射影空间中的闭复子流形是泛函$\mathcal {M}_{2p}$的临界点，称之为相对$2p$ -极小子流形，对所有$p$ .最后讨论了实空间形式中相对$2p$ -极小子流形与严格子流形的关系以及一个特殊的变分问题。
On quadratic orthogonal bisectional curvature
本文研究了在对分曲率上满足一定非负性条件的紧致K \ahler流形。在此条件下，我们证明了数量曲率是非负的，第一陈类是积极的假设局部不可约。在此条件下，我们还得到了泛覆盖的可能de Rham分解的部分分类
A minimal lamination of the interior of a positive cone with quadratic curvature blowup
我们用初等方法构造了R3中正锥内部的极小层。
About Some Quadratic Scalar Curvatures and the $h_{4}$ Yamabe Equation
这篇论文是根据2008年夏天在法国罗斯科夫举行的共形几何会议上的一次演讲撰写的。我们研究了在给定的维数至少为4的闭黎曼流形上存在具有常$h_4$曲率的共形度量的问题所引起的方程的某些方面。我们建立了第二Gauss-Bonnet曲率$h_4$与$σ_2$曲率之间的一个简单关系式，并研究了这两个二次曲率的一些正性性质.我们使用不同的二次曲率来刻画空间型、爱因斯坦度量和共形平坦度量。在附录中，我们介绍了牛顿变换的自然推广，并利用相应的牛顿恒等式得到了所有Gauss-Bonnet曲率的Avez型公式。
Some rigidity results on critical metrics for quadratic functionals
本文证明了单位体积黎曼度量空间上二次曲率泛函的临界度量的刚性结果，涉及Ricci和数量曲率.众所周知，爱因斯坦度量总是临界点。本文的目的是证明，在某些曲率条件下，部分逆是正确的。特别地，对于一类二次曲率泛函，我们证明了每一个具有非负截面曲率的临界度量必是Einstein的。
Kähler C-spaces and quadratic bisectional curvature
本文给出了具有$b_2=1$的不可约Kähler C-空间具有非负或正二次对分曲率的充要条件，并假定该空间不是Hermite对称的.在五个例外李群$E_6, E_7, E_8, F_4, G_2$的情况下，用MAPLE程序包进行了计算.这一结果与李五征的两种学说有关。
Uniqueness of critical metrics for a quadratic curvature functional
研究二次曲率泛函临界度量的唯一性，证明在维数 $n\geq 10$ 时有限能量的完备临界度量必标量平坦，即为全局极小值。
A variational proof for the existence of a conformal metric with preassigned negative Gaussian curvature for compact Riemann surfaces of genus $>1$
给定一个光滑函数$K <0$，我们证明了Berger，Kazhdan等人的一个结果：对于亏格为$g>1$的紧致Riemann曲面，在每一个共形类中都存在一个度量，使该函数达到其高斯曲率.我们这样做，通过最小化一个适当的功能使用初等分析。特别是当$K$为负常数时，这为亏格为$g >1$的紧致Riemann曲面的一致化定理提供了一个初等证明.
A Variational Characterization of Positive Scalar Curvature Kähler metrics
该研究在紧凯勒流形上引入规定标量曲率测度方程，证明正标量曲率凯勒度量存在性等四个命题等价，还得出正维紧光滑环面凯勒流形的凯勒类必含对应不变度量，且方程有唯一模常数的光滑解。
Conformally variational Riemannian invariants (CVIs), such as the scalar curvature, are homogeneous scalar invariants which arise as the gradient of a Riemannian functional. We establish a wide range of stability and rigidity results involving CVIs, generalizing many such results for the scalar curvature.
Examples of open manifolds with almost quadratic volume growth and infinite Betti numbers
本文构造了具有正Ricci曲率、截面曲率下有界和无限Betti数的开流形示例，其体积增长可任意接近二次，并探讨了此类流形在至多二次体积增长下是否必须具有有限拓扑类型的问题。
We use variational methods and a modified curvature flow to give an alternative proof of the existence of a self-shrinking torus under mean curvature flow. As a consequence of the proof, we establish an upper bound for the weighted energy of our shrinking doughnuts.
Quadratic Scalar Curvature Decay and Uniform Positivity in Dimensions Four through Seven
针对四维至七维满足渐近二次标量曲率系数大于(n-1)/n的正标量曲率完备非紧黎曼流形，证明其存在标量曲率至少为1的完备光滑度量，证实了Gromov临界衰减速率猜想的对应部分。
Variational characterizations of the total scalar curvature and eigenvalues of the Laplacian
对于数量曲率函数的线性化$s_g'$的对偶算子$s_g'^*$，众所周知，如果$\ker s_g'^*\neq 0$，则$s_g$是非负常数。特别地，如果Ricci曲率不是平坦的，则$ {s_g}/(n-1)$是度量$g$的拉普拉斯算子的本征值。本文对空间$\ker s_g'^*$进行了变分刻画。为了完成这个任务，我们引入了一个四阶椭圆微分算子$\mathcal A$和一个相关的几何不变$ν$。我们证明了$ν$为零当且仅当$\ker s_g'^* \ne 0$，并且如果拉普拉斯算子的第一特征值大于其数量曲率，则$ν$为正，并且$\ker s_g'^*= 0$。此外，我们还计算了在$\ker s_g'^* = 0$的情况下$ν$的下界。证明了若存在$\mathcal A$ -超调和函数且Ricci曲率有下界，则Laplace算子的第一非零特征值有上界.
A variational characterization of flat spaces in dimension three
在这个简短的说明中，我们证明了，在三维，平坦的度量是唯一的完整的度量与非负的数量曲率是至关重要的$σ_{2}$ -曲率功能。
In this paper, close surfaces are considered in 3-dimensional harmonic conformally flat space in point of the variation. It is shown that if the conformal vector field be tangent to surface and the sign of the mean curvature does not change then surface is minimal. Also, it is determined that the critical point of mean curvature functional is homeomorphic to the sphere. Furthermore, the Euler-Lagrange equations associated to the mean curvature and Willmore functionals are determined.
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。
New Rigidity Results for Critical Metrics of Some Quadratic Curvature Functionals
本文研究闭光滑流形上二次曲率泛函 $\mathfrak{F}_{t}$ 的临界度量，在截面曲率拼挤条件下证明新刚性，并推出四维 Bach 平坦常标量曲率度量的 Einstein 性与标准分类。
We study the supremum of the total mean curvature on the boundary of compact, mean-convex 3-manifolds with nonnegative scalar curvature, and a prescribed boundary metric. We establish an additivity property for this supremum and exhibit rigidity for maximizers assuming the supremum is attained. When the boundary consists of 2-spheres, we demonstrate that the finiteness of the supremum follows from the previous work of Shi-Tam and Wang-Yau on the quasi-local mass problem in general relativity. In turn, we define a variational analog of Brown-York quasi-local mass without assuming that the bound…
Critical metrics for the quadratic curvature functional on complete four-dimensional manifolds
本文研究了四维完备流形上曲率泛函的临界度量，证明其要么为爱因斯坦流形，要么局部等距于常曲率积空间。
On the variational properties of the prescribed Ricci curvature functional
本文研究齐次度量的预定 Ricci 曲率问题，通过考察标量曲率泛函的全局性质，为一般齐次空间给出了该泛函存在全局最大值的条件，并通过具体例子加以说明。
We investigate the $L^p$-boundness of the Riesz transform on Riemannian manifolds whose Ricci curvature has quadratic decay. Two criteria for the $L^p$-unboundness of the Riesz transform are given. We recover known results about manifolds that are Euclidean or conical at infinity.
Complete 3-manifolds of positive scalar curvature with quadratic decay
本文证明具有正数量曲率且无穷远处次二次衰减的完备可定向3-流形可分解为球面流形与S²×S¹的连通和，推广了Gromov-Wang定理并部分回答了Gromov猜想，且该衰减速率是最优的。
Rigidity and stability of Einstein metrics for quadratic curvature functionals
研究了黎曼度量空间上二次曲率泛函的临界点的刚性和稳定性。我们表明，它是可能的“衡量”的欧拉-拉格朗日方程，在一个自伴的方式，成为椭圆形。Fredholm理论可以用来描述临界度量的模空间的局部性质。我们证明了一些紧凑的例子是无穷小刚性的，因此，是孤立的临界点的空间中的单位体积黎曼度量。然后，我们给出了严格的局部极小（最多的同构和缩放）的关键指标的例子。一个推论是一个当地的“反向主教的不等式”等指标。具体地，在满足$Ric(g) \leq Ric(g_S)$的轮度量$(S^n,g_S)$的$C^{2,α}$邻域中的任何度量$g$具有体积$Vol(g) \geq Vol(g_S)$，等式成立当且仅当$g$与$g_S$等距。
In this paper we prove new rigidity results for complete, possibly non-compact, critical metrics of the quadratic curvature functionals $\mathfrak{F}^{2}_t = \int |\operatorname{Ric}_g|^{2} dV_g + t \int R^{2}_g dV_g$, $t\in\mathbb{R}$, and $\mathfrak{S}^2 = \int R_g^{2} dV_g$. We show that (i) flat surfaces are the only critical points of $\mathfrak{S}^2$, (ii) flat three-dimensional manifolds are the only critical points of $\mathfrak{F}^{2}_t$ for every $t>-\frac{1}{3}$, (iii) three-dimensional scalar flat manifolds are the only critical points of $\mathfrak{S}^2$ with finite energy and (iv…
We exhibit a curious link between the Quadratic Orthogonal Bisectional Curvature, combinatorics, and distance geometry. The Weitzenböck curvature operator, acting on real (1,1)--forms, is realized as the Dirichlet energy of a finite graph, weighted by a matrix of the curvature. These results also illuminate the difference in the nature of the Quadratic Orthogonal Bisectional Curvature and the Real Bisectional Curvature.
In this paper, we study the prescribed $Q$-curvature problem on closed four-dimensional Riemannian manifolds when the total integral of the $Q$-curvature is a positive integer multiple of the one of the four-dimensional round sphere. This problem has a variational structure with a lack of compactness. Using some topological tools of the theory of "critical points at infinity" combined with a refined blow-up analysis and some dynamical arguments, we identify the accumulations points of all noncompact flow lines of a pseudogradient flow, the so called critical points at infinity of the associate…
Uniform positive scalar curvature from supercritical quadratic decay
本文证明在超临界二次衰减条件下，完备非紧黎曼流形可配备一致正数量曲率度量，并将结果推广至所有维度且无需可定向性假设。
Non-negatively curved Kähler manifolds with average quadratic curvature decay
设$(M, g)$是一个完备的非紧Kähler流形，它具有非负的有界全纯双截曲率.推广我们在\cite{CT3}中发展的技巧，我们证明了$M$的泛覆盖$\wt M$是$\ce^n$的双全纯，只要$(M, g)$具有平均二次曲率衰减，或者$M$支持具有非负和一致有界全纯双截曲率的Kähler-里奇流的永恒解.我们还分类某些局部限制所产生的Kähler-里奇流的注入半径的统一估计的情况下。
Variational status of a class of fully nonlinear curvature prescription problems
通过保角变换，Schouten张量$P$的k阶初等对称多项式为常数，这是Yamabe问题的推广。在紧致的黎曼n-流形上，我们证明了，对于k在3和n之间（包括3和n），这个处方方程是某个作用量的欧拉-拉格朗日方程当且仅当结构是局部共形平坦的。
In this note we continue the analysis of metric measure space with variable ricci curvature bounds. First, we study $(κ,N)$-convex functions on metric spaces where $κ$ is a lower semi-continuous function, and gradient flow curves in the sense of a new evolution variational inequality that captures the information that is provided by $κ$. Then, in the spirit of previous work by Erbar, Kuwada and Sturm \cite{erbarkuwadasturm} we introduce an entropic curvature-dimension condition $CD^e(κ,N)$ for metric measure spaces and lower semi-continuous $κ$. This condition is stable with respect to Gromov …
Surfaces with parallel mean curvature vector in complex space forms
本文讨论了定义在任意维复空间型的具有平行中曲率向量的曲面上的二次型，证明了它的$(2,0)$ -部分是全纯的。当周围空间的复维数等于$2$时，我们定义了一个具有相同性质的二次型，然后确定了具有平行中曲率向量的曲面，在这些曲面上，它们的$(2,0)$ -部分都为零。给出了一个余维约化定理，证明了具有平行中曲率向量的$2$ -球面的不存在性。
We study the spectrum of phase transitions with prescribed mean curvature in Riemannian manifolds. These phase transitions are solutions to an inhomogeneous semilinear elliptic PDE that give rise to diffuse objects (varifolds) that limit to hypersurfaces, possibly with singularities, whose mean curvature is determined by the "prescribed mean curvature" function and the limiting multiplicity. We establish upper bounds for the eigenvalues of the diffuse problem, as well as the more subtle lower bounds when the diffuse problem converges with multiplicity one. For the latter, we also establish asy…
Optimal decay constant for complete manifolds of positive scalar curvature with quadratic decay
本文证明了正数量曲率完备3维流形在二次衰减常数大于2/3时可分解为球面流形与S²×S¹的连通和，并推广到4、5维得到拓扑障碍，部分回答了Gromov猜想。
On Ricci curvature and volume growth in dimension three
证明了R^3上的完备度量减去一个具有非负里奇曲率和二次Ricci曲率衰减的球，具有立方体积增长。
Curvature explosion in quotients and applications
证明了变分完全群作用的商空间是良黎曼轨道折叠。将所得结果推广到不含水平共轭点的奇异黎曼叶理。
The Ambient Obstruction Tensor and Q-Curvature
证明了Branson的Q曲率积分的变分导数是Fürkman-Graham的环境阻塞张量。建立了曲率中模二次项和高次项不可约共形不变张量的分类。
We study the resonant prescribed T-curvature problem on a compact 4-dimensional Riemannian manifold with boundary. We derive sharp energy and gradient estimates of the associated Euler-Lagrange functional to characterize the critical points at infinity of the associated variational problem under a non-degeneracy on a naturally associated Hamiltonian function. Using this, we derive a Morse type lemma around the critical points at infinity. Using the Morse lemma at infinity, we prove new existence results of Morse theoretical type. Combining the Morse lemma at infinity and the Liouville version …
Generalized Bianchi identities in gauge-natural field theories and the curvature of variational principles
借助于Noether第二定理，我们将规范自然丛上的拉格朗日场论的广义Bianchi恒等式与相应的规范自然Jacobi态射的核联系起来.规范自然变分原理的曲率的一个合适的定义，因此可以制定的哈密尔顿连接规范地与一个广义拉格朗日通过收缩场方程。
Variational Effect of Boundary Mean Curvature on ADM Mass in General Relativity
本文推广了\cite{Miao}的思想和技巧，研究了边界几何对渐近平坦流形ADM质量的变分效应。证明了对于拟凸边界的有界黎曼区域的Lipschitz渐近平坦度量扩张，如果扩张的边界平均曲率受控于但不恒等于给定区域所确定的边界平均曲率，则可以在提高边界平均曲率的同时降低其ADM质量.因此，我们的分析意味着，对于一个区域的拟凸边界，几何边界条件成立Bartnik的最小质量扩张猜想\cite{Bartnik_energy}。
Variational analysis of a mean curvature flow action functional
我们考虑约化的Allen-Cahn作用泛函，它表现为Allen-Cahn作用泛函的尖锐界面极限，可以理解为随机扰动中曲率流的形式作用泛函.对于广义超曲面的适当演化，该泛函由中曲率和速度向量的平方和组成，在时间和空间上积分。对于给定的初始和最终条件，我们研究相应的行动最小化问题。我们给出了一个推广的形式，并证明了作用泛函的紧性和下半连续性。此外，我们还刻画了欧拉-拉格朗日方程的光滑驻点，并研究了守恒量。最后，我们提出了一个明确的例子，并考虑同心球作为初始和最终的数据和特征的依赖于给定的时间跨度的最小旋转对称连接的属性。
This is our second paper in a series to study gravitational instantons, i.e. complete hyperkäler 4-manifolds with faster than quadratic curvature decay. We prove two main theorems: 1.The asymptotic rate of gravitational instantons to the standard models can be improved automatically. 2.Any ALF-D_k gravitational instanton must be the Cherkis-Hitchin-Ivanov-Kapustin-Lindström-Roček metric.
Variational Properties of the Gauss-Bonnet Curvatures
$2k$阶的高斯-博内特曲率是高斯-博内特被积函数在维度$2k$中的更高维度的推广，因为通常的数量曲率推广了二维高斯-博内特被积函数。在本文中，我们评估的一阶变分的积分，这些曲率视为泛函的空间上的所有黎曼度量的流形上正在考虑。这个导数的一个重要性质是它只依赖于曲率张量而不依赖于它的共变导数。我们表明，这个功能的临界点一旦限制到单位体积的度量是广义爱因斯坦度量，一旦限制到逐点共形类的度量是度量与常数高斯-邦纳曲率。
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