C. Robin Graham, Kengo Hirachi
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.
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Extended Solitons of the Ambient Obstruction Flow
本文证明闭环境障碍孤子必为环境障碍平坦且标量曲率恒定，并推广至扩展孤子方程，指出紧致流形上环境障碍流无共形不动点，同时推广了Bourguignon-Ezin恒等式。
We examine homogeneous solitons of the ambient obstruction flow and, in particular, prove that any compact ambient obstruction soliton with constant scalar curvature is trivial. Focusing on dimension 4, we show that any homogeneous gradient Bach soliton that is steady must be Bach flat, and that the only non-Bach-flat shrinking gradient solitons are product metrics on $\mathbb{R}^2\times S^2$ and $\mathbb{R}^2 \times H^2$. We also construct a non-Bach-flat expanding homogeneous gradient Bach soliton. We also establish a number of results for solitons to the geometric flow by a general tensor $…
For a conformal manifold, we describe a new relation between the ambient obstruction tensor of Fefferman and Graham and the holonomy of the normal conformal Cartan connection. This relation allows us to prove several results on the vanishing and the rank of the obstruction tensor, for example for conformal structures admitting twistor spinors or normal conformal Killing forms. As our main tool we introduce the notion of a conformal holonomy distribution and show that its integrability is closely related to the exceptional conformal structures in dimensions five and six that were found by Nurow…
Some obstructions to positive scalar curvature on a noncompact manifold
本文研究非紧流形上存在完备正数量曲率度量的障碍，并分别对有限体积与无限体积情形给出判定条件。
We provide integral curvature bounds for compact Riemannian manifolds that allow isometric immersions into a Euclidean space with low codimension in terms of the Betti numbers.
Torsion Obstructions to Positive Scalar Curvature
本文研究由闭光滑流形基本群整同调中的挠类引起的正数量曲率度量存在性的阻碍，并构造了不承认正数量曲率度量但其笛卡尔积承认此类度量的流形新例子。
Motivated to study the geometry of the exotic spheres constructed in [5], we derive a necessary condition for non-negative sectional curvature in certain total spaces of Riemannian submersions with totally geodesic fibers. In particular, we prove that the bundles in [5] and [1] have sections of negative curvature.
The (Fefferman-Graham) ambient obstruction tensor is a conformally invariant symmetric trace-free 2-tensor on even-dimensional Riemannian and pseudo-Riemannian manifolds. The conformal deformation complex is a differential complex related to infinitesimal deformations of conformal structure. We construct a conformally invariant differential operator on algebraic Weyl tensors that gives special curved analogues of certain operators related to the deformation complex and that, upon application to the Weyl curvature, yields the obstruction tensor. This new definition of the obstruction tensor lea…
We show that the indices of certain twisted Dirac operators vanish on a $Spin$-manifold $M$ of positive sectional curvature if the symmetry rank of $M$ is $\geq 2$ or if the symmetry rank is one and $M$ is two connected. We also give examples of simply connected manifolds of positive Ricci curvature which do not admit a metric of positive sectional curvature and positive symmetry rank.
Positive Scalar Curvature Obstructions via Singular Dimension Descent
本文发展了Schoen-Yau型奇异维度下降法，用于任意维度的正数量曲率障碍研究，证明了可放大流形上的正数量曲率障碍，并建立了相应的立方宽度不等式和双系统估计。
We establish new obstruction results to the existence of Riemannian metrics on tori satisfying mixed bounds on both their sectional and Ricci curvatures. More precisely, from Lohkamp's theorem, every torus of dimension at least three admits Riemannian metrics with negative Ricci curvature. We show that the sectional curvature of these metrics cannot be bounded from above by an arbitrarily small positive constant. In particular, if the Ricci curvature of a Riemannian torus is negative, bounded away from zero, then there exist some planar directions in this torus where the sectional curvature is…
We establish fundamental results for a parabolic flow of Riemannian metrics introduced by Bahuaud-Helliwell in arXiv:1010:4287v1 which is based on the Fefferman-Graham ambient obstruction tensor. First, we obtain local $L^2$ smoothing estimates for the curvature tensor and use them to prove pointwise smoothing estimates for the curvature tensor. We use the pointwise smoothing estimates to show that the curvature must blow up for a finite time singular solution. We also use the pointwise smoothing estimates to prove a compactness theorem for a sequence of solutions with bounded $C^0$ curvature …
K. Grove, L. Verdiani, B. Wilking and W. Ziller gave the first examples of cohomogeneity one manifolds which do not carry invariant metrics with non-negative sectional curvatures. In this paper we generalize their results to a larger family. We also classified all class one representations for a pair (G;H) with G/H some sphere, which are used to construct the examples.
The classical theory of $G$-structures, which include almost-complex structures, explains the relationship between the curvature of compatible connections and integrability. This note is an effort to understand how the curvature of Riemannian metrics can obstruct the integrability of almost-complex structures. It is shown that certain special complex structures cannot coexist with non-flat constant curvature metrics, and a formal variational realization of these structures is provided. The approach followed here is direct, meaning that it bypasses the classical theory. The idea is to find obst…
We study topological obstructions to the existence of a Riemannian metric on manifolds with boundary such that the scalar curvature is non-negative and the boundary is mean convex. We construct many compact manifolds with boundary which admit no Riemannian metric with non-negative scalar curvature and mean convex boundary. For example, we show that the manifold $(T^{n-2}\times Σ)\# N$, where $Σ$ is a compact, connected and orientable surface which is not a disk or a cylinder and $N$ is a closed $n$-dimensional manifold, does not admit a metric of non-negative scalar curvature and mean convex b…
In contrast to the homogeneous case, we show that there are compact cohomogeneity one manifolds, that do not support invariant metrics of non-negative sectional curvature. In fact we exhibit infinite families of such manifolds including the exotic Kervaire spheres. Such examples exist for any codimension of the singular orbits except for the case where both are equal to two, where existence of non-negatively curved metrics is known.
We prove a Pohozaev type identity for non-linear eigenvalue equations of the Dirac operator on Riemannian spin manifolds with boundary. As an application, we obtain that the mean curvature H of a conformal immersion S^{n}-> R^{n+1} satisfies $\int \partial_X H=0$ where X is a conformal vector field on S^{n} and where the integration is carried out with respect to the Euclidean volume measure of the image.<BR> This identity is analogous to the Kazdan-Warner obstruction that appears in the problem of prescribing the scalar curvature on S^{n} inside the standard conformal class.
We provide a quantitative obstruction to collapsing surfaces of genus at least 2 under a lower curvature bound and an upper diameter bound. Keywords: curvature; diameter; volume; filling radius; systole; Gromov-Hausdorff distance
Partial Scalar Curvatures and Topological Obstructions for Submanifolds
该研究引入介于最小Ricci曲率与归一化标量曲率之间的内蕴曲率ρₖ，推导空间形式中子流形的相关不等式与基于Betti数的拓扑障碍，并构造出第一Betti数任意大的𝕊⁶中三维极小Wintgen理想子流形验证k=n时障碍不成立。
A Gauss-Bonnet-Chern type obstruction for Killing vector fields on Lorentzian manifolds
本文提出了一种新的曲率障碍，利用类时（零）截面曲率刻画偶数维（奇数维）Lorentzian 流形上 Killing 或同位向量场的存在性，并将紧致情形下的 Gauss-Bonnet-Chern 障碍推广至非零类时截面曲率。
We give a geometric obstruction to the non-negativity of the sectional curvature in the total spaces of certain Riemannian submersions with totally geodesic fibers; applications of this obstruction to several examples are given.
We find new obstructions to the existence of complete Riemannian metric of nonnegative sectional curvature on manifolds with infinite fundamental groups. In particular, we construct many examples of vector bundles whose total spaces admit no nonnegatively curved metric.
We first show that existence results due to Kazdan-Warner and Cruz-Vitório can be extended to the category of manifolds with an isolated conical singularity. More precisely, we check that, under suitable conditions on the link manifold, any bounded and smooth function which is negative somewhere is the scalar curvature of some conical metric (with the boundary being minimal whenever it is non-empty). By way of comparison, we complement this analysis by indicating how index theory, as developed by Albin-Gell-Redman, may be used to transfer to this conical setting some of the classical obstructi…
Applying a local Gauss-Bonnet formula for closed subanalytic sets to the complex analytic case, we obtain characterizations of the Euler obstruction of a complex analytic germ in terms of the Lipschitz-Killing curvatures and the Chern forms of its regular part. We also prove analogous results for the global Euler obstruction. As a corollary, we give a positive answer to a question of Fu on the Euler obstruction and the Gauss-Bonnet measure.
Compactness of capillary hypersurfaces with mean curvature prescribed by ambient functions
本文证明了由环境函数规定平均曲率的毛细超曲面的紧致性定理，推广了Schätzle和Bellettini的结果，并扩展了曲率varifold定义至有向积分varifold，同时讨论了边界平均曲率规定情形。
We give a geometric derivation of Branson's Q-curvature in terms of the ambient metric associated with conformal structures; it naturally follows from the ambient metric construction of conformally invariant operators and can be applied to a large class of invariant operators. This procedure can be also applied to CR geometry and gives a CR analog of the Q-curvature; it then turns out that the Q-curvature gives the coefficient of the logarithmic singularity of the Szego kernel of 3-dimensional CR manifolds.
The behavior under conformal change of the renormalized volume coefficients associated to a pseudo-Riemannian metric is investigated. It is shown that they define second order fully nonlinear operators in the conformal factor whose algebraic structure is elucidated via the introduction of "extended obstruction tensors". These together with the Schouten tensor constitute building blocks for the coefficients in the ambient metric expansion. The renormalized volume coefficients have recently been considered by Chang-Fang motivated by comparison with the elementary symmetric functions of the eigen…
Existence and obstructions for the curvature on compact manifolds with boundary
本文研究带边紧流形上指定曲率函数的存在性问题，证明Gauss-Bonnet符号条件是共形等价度量实现测地曲率或高斯曲率的充要条件，并给出依赖Euler示性数的存在性与不存在性新结果。
In this paper we provide an extension to the Jellett-Minkowski's formula for immersed submanifolds into ambient manifolds which possesses a pole and radial curvatures bounded from above or below by the radial sectional curvatures of a rotationally symmetric model space. Using this Jellett-Minkowski's generalized formula we can focus on several isoperimetric problems. More precisely, on lower bounds for isoperimetric quotients of any precompact domain with smooth boundary, or on the isoperimetric profile of the submanifold and its modified volume. In the particular case of a model space with st…
In the article, we generalize some recent results of Colding and Minicozzi on generic singularities of mean curvature flow to curved ambient spaces. To do so, we make use of a weighted monotonicity formula to derive an "almost monotonicity" for the entropy upon embedding into $\R^\ell$. We are also lead to study the continuity of the entropy functional in certain cases.
Obstructions to prescribed Q-curvature of complete conformal metrics on $\mathbb{R}^n$
本文研究了 $\mathbb{R}^n$ 上完备共形度量的预定 Q-曲率问题，给出了一个 Bonnet-Mayer 型障碍定理，并揭示了预定函数衰减速率带来的其他障碍。
We extend the notions of CR GJMS operators and Q-curvature to the case of partially integrable CR structures. The total integral of the CR Q-curvature turns out to be a global invariant of compact nondegenerate partially integrable CR manifolds equipped with an orientation of the bundle of contact forms, which is nontrivial in dimension at least five. It is shown that its variation is given by the curvature-type quantity called the CR obstruction tensor, which is introduced in the author's previous work. Moreover, we consider the linearized CR obstruction operator. Based on a scattering-theore…
Block and Weinberger show that an arithmetic manifold can be endowed with a positive scalar curvature metric if and only if its $\rationals$-rank exceeds 2. We show in this article that these metrics are never in the same coarse class as the natural metric inherited from the base Lie group. Furthering the coarse $C^\ast$-algebraic methods of Roe, we find a nonzero Dirac obstruction in the $K$-theory of a particular operator algebra which encodes information about the quasi-isometry type of the manifold as well as its local geometry.
We prove integral curvature bounds in terms of the Betti numbers for compact submanifolds of the Euclidean space with low codimension. As an application, we obtain topological obstructions for $δ$-pinched immersions. Furthermore, we obtain intrinsic obstructions for minimal submanifolds in spheres with pinched second fundamental form.
Backward Uniqueness of Extrinsic Geometric Flow in general ambient manifolds
本文证明了外蕴几何流在一般 ambient 流形中的逆唯一性定理，适用于多种几何流类型。
We establish a link between rational homotopy theory and the problem which vector bundles admit complete Riemannian metric of nonnegative sectional curvature. As an application, we show for a large class of simply-connected nonnegatively curved manifolds that, if C lies in the class and T is a torus of positive dimension, then "most" vector bundles over the product of C and T admit no complete nonnegatively curved metric.
Lectures on Immersions with Controlled Curvatures
探讨黎曼流形间具“小”曲率浸入的构造方法，并指出此类浸入存在的障碍。
Twisted $S^1$ stability and positive scalar curvature obstruction on fiber bundles
本文研究纤维丛上的正标量曲率障碍，在纤维不可压缩或同伦非平凡条件下证明多种纤维丛无 PSC 度量，并给出 $S^1$ 丛的 PSC 稳定性等价条件，部分回答 Gromov 问题并推广前人结果。
Mean curvature flow into an ambient Riemannian manifold evolving by Ricci flow coupled with harmonic map heat flow
本文研究在Ricci流与调和映射热流耦合演化的带边流形中，平均曲率流的变分、Harnack不等式、Huisken单调性公式及孤子构造与刻画。
PSC Kähler cones of minimal Kähler surfaces and a birational obstruction
本文刻画了极小紧Kähler曲面的正标量曲率Kähler锥，并证明在双有理态射到斜率不稳定规则曲面时存在无正标量曲率度量的Kähler类，给出双有理障碍。
The scalar curvature in wedge spaces: existence and obstructions
本文研究具有单一奇异层的楔形空间中不完备楔形度量的数量曲率，利用 Yamabe、椭圆和指标理论，在正性假设下给出此类度量的存在性与阻碍结果，并推广了先前关于孤立锥奇点的工作。
Examples of almost-positively and quasi-positively curved spaces of the form M=H((G,h)xF) were discovered recently. Here, h is a left-invariant metric on a compact Lie group G, F is a compact Riemannian manifold on which the subgroup H of G acts isometrically on the left, and M is the orbit space of the diagonal left action of H on (G,h)xF with the induced Riemannian submersion metric. We prove that no new examples of strictly positive sectional curvature exist in this class of metrics. This result generalizes the case F={point} proven by Geroch.
For a manifold with an affine connection, we prove formulas which infinitesimally quantify the gap in a certain naturally defined open geodesic quadrilateral associated to a pair of tangent vectors $u$, $v$ at a point of the manifold. We show that the 1st order infinitesimal obstruction to the quadrilateral to close is always zero, the 2nd order infinitesimal obstruction to the quadrilateral to close is $-T(u,v)$ where $T$ is the torsion tensor of the connection, and if $T = 0$ then the 3rd order infinitesimal obstruction to the quadrilateral to close is $(1/2)R(u,v)(u+v)$ in terms of the curv…
Resolution of compact Einstein orbifolds in general dimensions
研究高维紧致爱因斯坦流形非塌缩序列的Gromov-Hausdorff极限为紧致爱因斯坦orbifold，证明其存在有限奇点，并给出超几何orbifold的负障碍物，简化了Ozuch四维工作中的技术难点。
Kazdan-Warner obstructions for a 4$th-$order boundary problem
本文研究通过共形变换设定非恒定内部Q曲率和边界T曲率的四阶边值问题，利用变分方法和共形向量场获得非平凡积分障碍。
Avoidance for Set-Theoretic Solutions of Mean-Curvature-Type Flows
本文研究任意光滑黎曼流形上带环境向量场的平均曲率型流，证明两个集合论下解在初始不相交且其中一个为紧时始终保持不相交，推广了此前仅限欧几里得空间无向量场情形的结果。
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.
Relative aspherical conjecture and higher codimensional obstruction to positive scalar curvature
研究相对非球面猜想，提出推广模型并引入球面宽度几何量，证明3-流形在余维3和4下对正数量曲率存在的障碍，扩展了维度5以内的非球面猜想。
Given a complete hypersurface isometrically immersed in an ambient manifold, in this paper we provide a lower bound for the norm of the mean curvature vector field of the immersion assuming that: 1) The ambient manifold admits a Killing submersion with unit-length Killing vector field. 2)The projection of the image of the immersion is bounded in the base manifold. 3)The hypersurface is stochastically complete, or the immersion is proper.
For an ancient solution of the mean curvature flow, we show that each time slice M_t is contained in an affine subspace with dimension bounded in terms of the density and the dimension of the evolving submanifold. Recall that an ancient solution is a family M_t that evolves under mean curvature flow for all negative time t.
We analyze the obstruction to metrics of positive scalar curvature within a given bounded distortion class of metrics. This obstruction lives in a non-Hausdorff cohomology group Poincare dual to the uniformly finite homology studied by Block and Weinberger. One of the applications is a converse to their theorem on infinite connected sums, giving a complete picture of when such manifolds have metrics of positive scalar curvature.
This is a survey of the current state of the question "Which closed connected manifolds of dimension $n\ge 5$ admit Riemannian metrics whose scalar curvature function is everywhere positive?" The introduction gives a brief overview of these results, while the body of the paper discusses the methods used in the proofs of these results. We mention the two flavors of topological obstructions to the existence of \pscm s: one is a consequence of the Weizenböck formula for the Dirac operator, the other is obtained by considering stable minimal hypersurfaces. We talk about geometric constructions of …
The Avoidance Principle for Noncompact Hypersurfaces Moving by Mean Curvature Flow
研究非紧超曲面在均曲率流中保持不相交的条件，证明在欧几里得空间或满足曲率有界条件的流形中，初始非零距离可保证曲面始终分离。
Obstructions to distinguished Riemannian metrics via Lorentzian geometry
本研究借助洛伦兹几何的对偶框架，通过平面波极限得到特殊黎曼度量的局部必要条件，并利用Wick旋转排除了一类偏离常曲率的紧致黎曼流形的存在性，为黎曼度量的曲率障碍研究提供了新方法。
On the Curvature and Topology of Compact Stationary Spacetimes
本文基于 Petersen 与 Wink 2021 年的结果，研究紧致洛伦兹流形在具有单位长度类时 Killing 向量场时其曲率与拓扑所受到的阻碍。
We show that under certain curvature conditions of the ambient space an entire Killing graph of constant mean curvature lying inside a slab must be a totally geodesic slice.
We study algebraic conditions on a group G under which every properly discontinuous, isometric G-action on a Hadamard manifold has a G-invariant Busemann function. For such G we prove the following structure theorem: every open complete nonpositively curved Riemannian K(G,1) manifold that is homotopy equivalent to a finite complex of codimension >2 is an open regular neighborhood of a subcomplex of the same codimension. In this setting we show that each tangential homotopy type contains infinitely many open K(G,1) manifolds that admit no complete nonpositively curved metric even though their u…
We show some computations related to the motion by mean curvature flow of a submanifold inside an ambient Riemannian manifold evolving by Ricci or backward Ricci flow. Special emphasis is given to the possible generalization of Huisken's monotonicity formula and its connection with the validity of some Li--Yau--Hamilton differential Harnack--type inequalities in a moving Riemannian manifold.
In this paper we obtain rigidity results and obstructions on the topology at infinity of translating solitons of the mean curvature flow in the Euclidean space. Our approach relies on the theory of f-minimal hypersurfaces.
Conformally compact metrics and the Lovelock tensors
本文研究满足推广Einstein方程的Lovelock方程的共形紧度量，证明其容许多重齐次展开以实现Fefferman-Graham展开，并在偶数维中识别出光滑性边界障碍，同时构造了奇异Yamabe-(2q)问题的形式解及自旋流形填充的指标障碍。
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