John Lott, Zhongmin Shen
We show that there are topological obstructions for a noncompact manifold to admit a Riemannian metric with quadratic curvature decay and a volume growth which is slower than that of Euclidean space of the same dimension.
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In a previous paper, we constructed complete manifolds of positive Ricci curvature with quadratically asymptotically nonnegative curvature and infinite topological type but dimension $\ge 6$. The purpose of the present paper is to use a different way to exhibit a family of complete $I$-dimensinal ($I\ge5$) Riemannian manifolds of positive Ricci curvature, quadratically asymptotically nonnegative sectional curvature, and certain infinite Betti number $b_j$ ($2\le j\le I-2$).
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。
We give sufficient conditions for a noncompact Riemannian manifold, which has quadratic curvature decay, to have finite topological type with ends that are cones over spherical space forms.
We give conditions which imply that a complete noncompact manifold with quadratic curvature decay has finite topological type. In particular, we find links between the topology of a manifold with quadractic curvature decay and some properties of the asymptotic cones of such a manifold.
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.
We construct a compact Kähler manifold of nonnegative quadratic bisectional curvature, which does not admit any Kähler metric of nonnegative orthogonal bisectional curvature. The manifold is a 7-dimensional Kähler C-space with second Betti number equal to 1, and its canonical metric is a Kähler-Einstein metric of positive scalar curvature
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 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.
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.
In this article we study compact K\ahler manifolds satisfying a certain nonnegativity condition on the bisectional curvature. Under this condition, we show that the scalar curvature is nonnegative and that the first Chern class is positive assuming local irreducibility. We also obtain a partial classification of possible de Rham decompositions of the universal cover under this condition
We use elementary methods to construct a minimal lamination of the interior of a positive cone in R3.
This is a paper based on a talk given at the conference on Conformal Geometry which held at Roscoff in France in the 2008 summer. We study some aspects of the equation arising from the problem of the existence on a given closed Riemannian manifold of dimension at leat 4, of a conformal metric with constant $h_4$ curvature. We establish a simple formula relating the second Gauss-Bonnet curvature $h_4$ to the $σ_2$ curvature and we study some positivity properties of these two quadratic curvatures. We use different quadratic curvatures to characterize space forms, Einstein metrics and conformall…
In this paper we prove rigidity results on critical metrics for quadratic curvature functionals, involving the Ricci and the scalar curvature, on the space of Riemannian metrics with unit volume. It is well-known that Einstein metrics are always critical points. The purpose of this article is to show that, under some curvature conditions, a partial converse is true. In particular, for a class of quadratic curvature functionals, we prove that every critical metric with non-negative sectional curvature must be Einstein.
In this article we give necessary and sufficient conditions for an irreducible Kähler C-space with $b_2=1$ to have nonnegative or positive quadratic bisectional curvature, assuming the space is not Hermitian symmetric. In the cases of the five exceptional Lie groups $E_6, E_7, E_8, F_4, G_2$, the computer package MAPLE is used to assist our calculations. The results are related to two conjectures of Li-Wu-Zheng.
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。
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。
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。
The goal of this paper is to investigate the topological structure of open simply-connected 3-manifolds whose scalar curvature has a slow decay at infinity. In particular, we show that the Whitehead manifold does not admit a complete metric, whose scalar curvature decays slowly, and in fact that any contractible complete 3-manifolds with such a metric is homeomorphic to $\mathbb{R}^{3}$. Furthermore, using this result, we prove that any open simply-connected 3-manifold $M$ with $π_{2}(M)=\mathbb{Z}$ and a complete metric as above, is homeomorphic to $\mathbb{S}^{2}\times \mathbb{R}$.
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。
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。
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。
We investigate rigidity and stability properties of critical points of quadratic curvature functionals on the space of Riemannian metrics. We show it is possible to "gauge" the Euler-Lagrange equations, in a self-adjoint fashion, to become elliptic. Fredholm theory may then be used to describe local properties of the moduli space of critical metrics. We show a number of compact examples are infinitesimally rigid, and consequently, are isolated critical points in the space of unit-volume Riemannian metrics. We then give examples of critical metrics which are strict local minimizers (up to diffe…
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.
We study the contraction of a convex immersed plane curve with speed (1/α)k^{α}, where αin(0,1] is a constant and show that, if the blow-up rate of the curvature is of type one, it will converge to a homothetic self-similar solution. We also discuss a special symmetric case of type two blow-up and show that it converges to a translational self-similar solution. In the case of curve shortening flow (i.e., when α=1), this translational self-similar solution is the familiar "Grim Reaper" (a terminology due to M. Grayson [GR]).
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。
Let $(M, g)$ be a complete non-compact Kähler manifold with non-negative and bounded holomorphic bisectional curvature. Extending our techniques developed in \cite{CT3}, we prove that the universal cover $\wt M$ of $M$ is biholomorphic to $\ce^n$ provided either that $(M, g)$ has average quadratic curvature decay, or $M$ supports an eternal solution to the Kähler-Ricci flow with non-negative and uniformly bounded holomorphic bisectional curvature. We also classify certain local limits arising from the Kähler-Ricci flow in the absence of uniform estimates on the injectivity radius.
We consider a quadratic form defined on the surfaces with parallel mean curvature vector of an any dimensional complex space form and prove that its $(2,0)$-part is holomorphic. When the complex dimension of the ambient space is equal to $2$ we define a second quadratic form with the same property and then determine those surfaces with parallel mean curvature vector on which the $(2,0)$-parts of both of them vanish. We also provide a reduction of codimension theorem and prove a non-existence result for $2$-spheres with parallel mean curvature vector.
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。
We prove that any complete metric on R^3 minus a ball with non-negative Ricci curvature and quadratic Ricci-curvature decay, has cubic volume growth.
Complete manifolds with nonnegative Ricci curvature and slow relative volume growth
研究非负瑞奇曲率完备非紧流形的相对体积增长特性，证明缓慢增长条件下基本群近似阿贝尔，且正瑞奇曲率时基本群有限。
In this paper the method of compensated compactness is applied to the problem of isometric immersion of a two dimensional Riemannian manifold with negative Gauss curvature into three dimensional Euclidean space. Previous applications of the method to this problem have required decay of order $t^{-4}$ in the Gauss curvature. Here we show that the decay of Hong $t^{-2-δ/2}$ where $δ\in(0,4)$ suffices.
Kähler manifolds and cross quadratic bisectional curvature
本文研究Kähler流形的CQB及其对偶曲率，证明正或非负曲率条件下的Fano性质，并通过Kähler C-空间例子揭示其与Betti数及对称性的关系。
There is a conjecture that a complete Riemannian 3-manifold with bounded sectional curvature, and pointwise pinched nonnegative Ricci curvature, must be flat or compact. We show that this is true when the negative part (if any) of the sectional curvature decays quadratically.
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.
It is shown that the variational derivative of the integral of Branson's Q-curvature is the ambient obstruction tensor of Fefferman-Graham. A classification of irreducible conformally invariant tensors modulo quadratic and higher degree terms in curvature is established.
In this paper, we study gravitational instantons (i.e., complete hyperkäler 4-manifolds with faster than quadratic curvature decay). We prove three main theorems: 1.Any gravitational instanton must have known end----ALE, ALF, ALG or ALH. 2.In ALG and ALH-non-splitting cases, it must be biholomorphic to a compact complex elliptic surface minus a divisor. Thus, we confirm a long-standing question of Yau in ALG and ALH cases. 3.In ALF-D_k case, it must have an O(4)-multiplet.
By establishing two general quadratic inequalities, we obtain some inequalities related to Ricci curvatures for Lagrangian submanifolds of K$\ddot{\mathrm{a}}$hler QCH-manifolds, which generalize some results for Lagrangian submanifolds of complex space forms.
We define holomorphic quadratic differentials for spacelike surfaces with constant mean curvature in the Lorentzian homogeneous spaces $\mathbb{L}(κ,τ)$ with isometry group of dimension 4, which are dual to the Abresch-Rosenberg differentials in the Riemannian counterparts $\mathbb{E}(κ,τ)$, and obtain some consequences. On the one hand, we give a very short proof of the Bernstein problem in Heisenberg space, and provide a geometric description of the family of entire graphs sharing the same differential in terms of a 2-parameter conformal deformation. On the other hand, we prove that entire m…
A Finsler space is called Ricci-quadratic if its Ricci curvature $Ric(x,y)$ is quadratic in $y$. It is called a Berwald space if its Chern connection defines a linear connection directly on the underlying manifold $M$. In this article, we prove that a homogeneous Randers space is Ricci-quadratic if and only if it is of Berwald type.
Ancient mean curvature flow asymptotic to a minimal quadratic cone
本文证明n≥5时，渐近于O(n)×O(n)对称西蒙斯锥且位于其一侧的平均曲率流古解在抛物区域内有唯一渐近行为，平均凸性假设下则完全唯一，且为Hardt-Simon叶状结构的定常流。
This is our third paper in a series on the gravitational instantons. In this paper, we classify ALG and ALH gravitational instantons. In ALG case, we extend Hein's construction slightly and show that it's the only ALG gravitational instanton. In ALH case, we prove a Torelli-type theorem.
Some model reduction techniques for multiple time-scale dynamical systems make use of the identification of low dimensional slow invariant attracting manifolds (SIAM) in order to reduce the dimensionality of the phase space by restriction to the slow flow. The focus of this work is on a proposition and discussion of a general viewpoint using differential geometric concepts for submanifolds to deal with slow invariant manifolds in an extended phase space. The motivation is a coordinate independent formulation of the manifold properties and its characterization problem treating the manifold as i…
The theory of slow invariant manifolds (SIMs) is the foundation of various model-order reduction techniques for dissipative dynamical systems with multiple time-scales, e.g. in chemical kinetic models. The construction of SIMs and many approximation methods exploit the restrictive requirement of an explicit time-scale separation parameter. Most of those methods are also not formulated covariantly, i.e. in terms of tensorial constructions. We propose an intrinsically coordinate-free differential geometric approximation criterion approximating normally attracting invariant manifolds (NAIMs). We …
On the Bach tensor and quadratic curvature functionals
本文研究黎曼四流形上依赖Bach张量梯度的二次泛函，证明闭流形上存在弱Bach平行度量，给出曲面乘积分类，构造新Bach平坦度量，并建立四维柱体与三维临界度量的等价性。
This paper concerns integral varifolds of arbitrary dimension in an open subset of Euclidean space satisfying integrability conditions on their first variation. Firstly, the study of pointwise power decay rates almost everywhere of the quadratic tilt-excess is completed by establishing the precise decay rate for two-dimensional integral varifolds of locally bounded first variation. In order to obtain the exact decay rate, a coercive estimate involving a height-excess quantity measured in Orlicz spaces is established. Moreover, counter-examples to pointwise power decay rates almost everywhere o…
We study the mean curvature flow of smooth $n$-dimensional compact submanifolds with quadratic pinching in a Riemannian manifold $\mathcal{N}^{n+m}$. Our main focus is on the case of high codimension, $m\geq 2$. We establish a codimension estimate that shows in regions of high curvature, the submanifold becomes approximately codimension one in a quantifiable way. This estimate enables us to prove at a singular time of the flow, there exists a rescaling that converges to a smooth codimension-one limiting flow in Euclidean space. Under a cylindrical type pinching, this limiting flow is weakly co…
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