Naichung Conan Leung, Tom Yau-heng Wan
We study the topology of a complete asymptotically hyperbolic Einstein manifold such that its conformal boundary has positive Yamabe invariant. We proved that all maps from such manifold into any nonpositively curved manifold are homotopically trivial. Our proof is based on a Bochner type argument on harmonic maps.
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We classify the harmonic morphisms with one-dimensional fibres (1) from real-analytic conformally-flat Riemannian manifolds of dimension at least four, and (2) between conformally-flat Riemannian manifolds of dimensions at least three.
We show that under some non-degeneracy assumption the only submersive harmonic morphism on a conformally flat $3-$sphere is the Hopf fibration. The proof involves an appropriate use the Chern-Simons functional.
Uniqueness of conformal-harmonic maps on locally conformally flat 4-manifolds
本文研究局部共形平坦4-流形到球面的共形调和映射，利用共形不变能量泛函的凸性证明了定量唯一性结果，并为此建立了流形上的二阶 Hardy 不等式。
The author has elsewhere given a complete classification of those compact oriented Einstein 4-manifolds on which the self-dual Weyl curvature is everywhere positive in the direction of some self-dual harmonic 2-form. In this article, similar results are obtained when the self-dual Weyl curvature is everywhere non-negative in the direction of a self-dual harmonic 2-form that is transverse to the zero section of the bundle of self-dual 2-forms. However, this transversality condition plays an essential role in the story; dropping it leads one into wildly different territory where entirely differe…
In this article, we study harmonic symmetries on the compact locally conformally Kähler manifold $M$ of $dim_{\mathbb{C}}=n$. The space of harmonic symmetries is a subspace of harmonic differential forms which defined by the kernel of a certain Laplacian-type operator $\square$. We observe that the spaces $\ker(\square)\capΩ^{l}=\{0\}$ for any $|l-n|\geq2$ and $\kerΔ_{\bar{\partial}}\cap P^{k,n-1-k}\cap\ker(i_{θ^{\sharp}})\cong\ker(\square^{k,n-1-k})$, $\kerΔ_{\bar{\partial}}\cap P^{k,n-k}\cong\ker(\square^{k,n-k})$. Furthermore, suppose that $M$ is a Vaisman manifold, we prove that (i) $α$ is…
In this paper we study the topology of conformally compact Einstein 4-manifolds. When the conformal infinity has positive Yamabe invariant and the renormalized volume is also positive we show that the conformally compact Einstein 4-manifold will have at most finite fundamental group. Under the further assumption that the renormalized volume is relatively large, we conclude that the conformally compact Einstein 4-manifold is diffeomorphic to $B^4$ and its conformal infinity is diffeomorphic to $S^3$.
In this paper, we establish some compactness results of conformally compact Einstein metrics on $4$-dimensional manifolds. Our results were proved under assumptions on the behavior of some local and non-local conformal invariants, on the compactness of the boundary metrics at the conformal infinity, and on the topology of the manifolds.
We study a characterization of 4-dimensional (not necessarily complete) gradient Ricci solitons $(M, g, f)$ which have harmonic Weyl curvature, i.e. $δW=0$. Roughly speaking, we prove that the soliton metric $g$ is locally isometric to one of the following four types: an Einstein metric, the product $ \mathbb{R}^2 \times N_λ$ of the Euclidean metric and a 2-d Riemannian manifold of constant curvature $λ \neq 0$, a certain singular metric and a locally conformally flat metric. The method here is motivated by Cao-Chen's works \cite{CC1, CC2} and Derdziński's study on Codazzi tensors \cite{De}. C…
We study conformal invariants that arise from functions in the nullspace of conformally covariant differential operators. The invariants include nodal sets and the topology of nodal domains of eigenfunctions in the kernel of GJMS operators. We establish that on any manifold of dimension $n\geq 3$, there exist many metrics for which our invariants are nontrivial. We discuss new applications to curvature prescription problems.
This paper focuses on the development of harmonic and Clifford analysis techniques in the context of some conformally flat manifolds that arise from factoring out a simply-connected domain from $R^n$ by special arithmetic subgroups of the conformal group. Our discussion encompasses in particular the Hopf manifold $S^1 \times S^{n-1}$, conformally flat cylinders and tori and some conformally flat manifolds of genus $g \ge 2$, such as $k$-handled tori and polycylinders. This paper provides a continuation as well as an extension of our previous two papers \cite{KraRyan1,KraRyan2}. In particular, …
First, we show that a warped product of a line and a fiber manifold is weakly conformally flat and quasi Einstein if and only if the fiber is Einstein. Next, we characterize and classify contact (in particular, $K$-contact) Riemannian manifold satisfying weakly (and doubly weakly) conformally flat and quasi-Einstein ($η$-Einstein) conditions. Finally, we provide local classification and characterization of a semi-Riemannian (including the 4-dimensional spacetime) with harmonic Weyl tensor and a non-homothetic conformal (including closed) vector field, in terms of Petrov types and Bach tensor.
A Willmore surface $y:M\rightarrow S^{n+2}$ has a natural harmonic oriented conformal Gauss map $Gr_y:M\rightarrow SO^{+}(1,n+3)/SO(1,3)\times SO(n)$, which maps each point $p\in M$ to its oriented mean curvature 2-sphere at $p$. An easy observation shows that all conformal Gauss maps of Willmore surfaces satisfy a restricted nilpotency condition which will be called "strongly conformally harmonic." The goal of this paper is to characterize those strongly conformally harmonic maps from a Riemann surface $M$ to $SO^+ (1, n + 3)/{SO^+(1, 3) \times SO(n) }$ which are the conformal Gauss maps of s…
In this paper, we focus on the geometry of compact conformally flat manifolds $(M^n,g)$ with positive scalar curvature. Schoen-Yau proved that its universal cover $(\widetilde{M^n},\tilde{g})$ is conformally embedded in $\mathbb{S}^n$ such that $M^n$ is a Kleinian manifold. Moreover, the limit set of the Kleinian group has Hausdorff dimension $<\frac{n-2}{2}$. If additionally we assume that the non-local curvature $Q_{2γ}\geq 0$ for some $1<γ<2$, the Hausdorff dimension of the limit set is less than or equal to $\frac{n-2γ}{2}$. If $Q_{2γ}>0$, then the above inequality is strict. Moreover, the…
We give conditions on the Lee vector field of an almost Hermitian manifold such that any holomorphic map from this manifold into a (1,2)-symplectic manifold must satisfy the fourth-order condition of being biharmonic, hence generalizing the Lichnerowicz theorem on harmonic maps. These third-order non-linear conditions are shown to greatly simplify on l.c.K. manifolds and construction methods and examples are given in all dimensions.
Conformal harmonic maps from a 4-dimensional conformal manifold to a Riemannian manifold are maps satisfying a certain conformally invariant fourth order equation. We prove a general existence result for conformal harmonic maps, analogous to the Eells-Sampson theorem for harmonic maps. The proof uses a geometric flow and relies on results of Gursky-Viaclovsky and Lamm.
Dimension of polynomial growth harmonic functions on locally conformally flat manifolds with nonnegative Ricci curvature
本文针对具非负Ricci曲率的局部共形平坦流形，证实丘成桐提出的关于多项式增长调和函数空间维数的第二个问题，即其维数不超过对应欧几里得空间的该空间维数的精确上界成立。
Locally conformally Kahler (LCK) manifolds with potential are those which admit a Kahler covering with a proper, automorphic Kaehler potential. Existence of a potential can be characterized cohomologically as a vanishing of a certain cohomology class, called the Bott-Chern class. Compact LCK manifolds with potential are stable at small deformations and admit holomorphic embeddings into Hopf manifolds. This class strictly includes the Vaisman manifolds. We show that every compact LCK manifold with potential can be deformed into a Vaisman manifold. Therefore, every such manifold is diffeomorphic…
We show that any polyhomogeneous asymptotically hyperbolic constant-mean-curvature solution to the vacuum Einstein constraint equations can be approximated, arbitrarily closely in Hölder norms determined by the physical metric, by shear-free smoothly conformally compact vacuum initial data.
Mass-capacity inequality modeled on conformally flat manifolds
针对旋量情形下带非负数量曲率的广义渐近平坦流形，建立质量-容量不等式，明确等号成立时流形的调和共形等价结构及对应子集的Hausdorff维数约束。
We say that a distribution is harmonic if it is harmonic when considered as a section of a Grassmann bundle. We find new examples of harmonic distributions and show nonexistense of harmonic distrubutions on some Riemannian manifolds by two different approaches. Firstly, we lift distributions to the second tangent bundle equipped with the Sasaki metric. Secondly, we deform conformally the metric on a base manifold.
We find geometric conditions on a Hermitian-Weyl manifold under which the complex structure is a pseudo-harmonic map in the sense of G. Kokarev \cite{K09} from the manifold into its twistor space. This is done under the assumption that the dimension of the manifold is four or the Hermitian-Weyl structure is locally conformally Kähler.
We prove that the dimension $h^{1,1}_{\overline\partial}$ of the space of Dolbeault harmonic $(1,1)$-forms is not necessarily always equal to $b^-$ on a compact almost complex 4-manifold endowed with an almost Hermitian metric which is not locally conformally almost Kähler. Indeed, we provide examples of non integrable, non locally conformally almost Kähler, almost Hermitian structures on compact 4-manifolds with $h^{1,1}_{\overline\partial}=b^-+1$. This answers to a question by Holt.
The Calderón problem for harmonic spinors
本文证明具有MIT边界条件的Dirac算子的边界共轭映射唯一确定紧致旋量流形的共形类，并推广到扭曲Dirac算子及实解析酉联络的局部规范等价性。
We give some rigidity theorems for an n$(\geq4)$-dimensional compact Riemannian manifold with harmonic Weyl curvature, positive scalar curvature and positive constant $σ_2$. Moreover, when $n=4,$ we prove that a 4-dimensional compact locally conformally flat Riemannian manifold with positive scalar curvature and positive constant $σ_2$ is isometric to a quotient of the round $\mathbb{S}^4$.
Asymptotic Dirichlet problem for harmonic maps and conformal geodesics
该研究利用从双曲平面到共形紧爱因斯坦流形的调和映射的渐近狄利克雷问题，对无穷远边界上的共形测地线进行全息刻画，其方法受Fine和Herfray重归一化面积最小化工作启发。
In this note we study the conformal metrics of constant $Q$ curvature on closed locally conformally flat manifolds. We prove that for a closed locally conformally flat manifold of dimension $n\geq 5$ and with Poincarë exponent less than $\frac {n-4}2$, the set of conformal metrics of positive constant $Q$ and positive scalar curvature is compact in the $C^\infty$ topology.
In this paper we study the existence and compactness of positive solutions to a family of conformally invariant equations on closed locally conformally flat manifolds. The family of conformally covariant operators $P_α$ were introduced via the scattering theory for Poincaré metrics associated with a conformal manifold $(M^n, [g])$. We prove that, on a closed and locally conformally flat manifold with Poincaré exponent less than $\frac {n-α}2$ for some $α\in [2, n)$, the set of positive smooth solutions to the equation $$ P_αu = u^\frac {n+α}{n-α} $$ is compact in the $C^\infty$ topology. There…
$\overline\partial$-Harmonic forms on $4$-dimensional almost-Hermitian manifolds
研究四维紧近复流形上埃尔米特度量的$\overline\partial$-调和(1,1)-形式维数，证明全局共形凯勒时维数为$b_-+1$，局部共形凯勒时维数为$b_-$，并给出Kodaira-Thurston流形上的例子表明该维数依赖于度量。
In this paper we introduce the notion of generalized quasi--Einstein manifold, that generalizes the concepts of Ricci soliton, Ricci almost soliton and quasi--Einstein manifolds. We prove that a complete generalized quasi--Einstein manifold with harmonic Weyl tensor and with zero radial Weyl curvature, is locally a warped product with $(n-1)$--dimensional Einstein fibers. In particular, this implies a local characterization for locally conformally flat gradient Ricci almost solitons, similar to that proved for gradient Ricci solitons.
We prove a conformally invariant estimate for the index of Schrödinger operators acting on vector bundles over four-manifolds, related to the classical Cwikel-Lieb-Rozenblum estimate. Applied to Yang-Mills connections we obtain a bound for the index in terms of its energy which is conformally invariant, and captures the sharp growth rate. Furthermore we derive an index estimate for Einstein metrics in terms of the topology and the Einstein-Hilbert energy. Lastly we derive conformally invariant estimates for the Betti numbers of an oriented four-manifold with positive scalar curvature.
The family of Willmore immersions from a Riemann surface into $S^{n+2}$ can be divided naturally into the subfamily of Willmore surfaces conformally equivalent to a minimal surface in $\R^{n+2}$ and those which are not conformally equivalent to a minimal surface in $\R^{n+2}$. On the level of their conformal Gauss maps into $Gr_{1,3}(\R^{1,n+3})=SO^+(1,n+3)/SO^+(1,3)\times SO(n)$ these two classes of Willmore immersions into $S^{n+2}$ correspond to conformally harmonic maps for which every image point, considered as a 4-dimensional Lorentzian subspace of $\R^{1,n+3}$, contains a fixed lightlik…
We are studying the harmonic and twistor equation on Lorentzian surfaces, that is a two dimensional orientable manifold with a metric of signature $(1,1)$. We will investigate the properties of the solutions of these equations and try to relate the conformal invariant dimension of the space of harmonic and twistor spinors to the natural conformal invariants given by the Lorentzian metric. We will introduce the notion of semi-conformally flat surfaces and establish a complete classification of the possible dimensions for this family.
Quasi-Einstein manifolds with Harmonic Weyl curvature
本文研究了具有调和韦伊尔曲率的准爱因斯坦流形，扩展了四维空间中的相关工作，并给出了新的非局部共形扁且非D-平坦的准爱因斯坦流形例子。
We consider harmonic immersions in $\R^{\N}$ of compact Riemann surfaces with finitely many punctures where the harmonic coordinate functions are given as real parts of meromorphic functions. We prove that such surfaces have finite total Gauss curvature. The contribution of each end is a multiple of $2π$, determined by the maximal pole order of the meromorphic functions. This generalizes the well known Gackstatter-Jorge-Meeks formula for minimal surfaces. The situation is complicated as the ends are generally not conformally equivalent to punctured disks, nor does the surface have limit tangen…
Topology of 3-manifolds with nonnegative scalar curvature and positive harmonic functions
研究具有非负标量曲率的完整3维流形，证明可收缩流形与R^3同胚，且允许此类度量的开放柄体 genus 至多为1。
We study complete minimal graphs in HxR, which take asymptotic boundary values plus and minus infinity on alternating sides of an ideal inscribed polygon Γ in H. We give necessary and sufficient conditions on the "lenghts" of the sides of the polygon (and all inscribed polygons in Γ) that ensure the existence of such a graph. We then apply this to construct entire minimal graphs in HxR that are conformally the complex plane C. The vertical projection of such a graph yields a harmonic diffeomorphism from C onto H, disproving a conjecture of Rick Schoen.
In this paper we prove that a complete, embedded minimal surface $M$ in $\mathbb{R}^3$ with finite topology and compact boundary (possibly empty) is conformally a compact Riemann surface $\overline{M}$ with boundary punctured in a finite number of interior points and that $M$ can be represented in terms of meromorphic data on its conformal completion $\overline{M}$. In particular, we demonstrate that $M$ is a minimal surface of finite type and describe how this property permits a classification of the asymptotic behavior of $M$.
This work deals with the conformal transformations in six-dimensional spinorial formalism. Several conformally invariant equations are obtained and their geometrical interpretation are worked out. Finally, the integrability conditions for some of these equations are established. Moreover, in the course of the article, some useful identities involving the curvature of the spinorial connection are attained and a digression about harmonic forms and more general massless fields is made.
Manifolds with harmonic Weyl curvature and curvature operator of the second kind
该研究证明了高维流形在特定曲率条件下的分类，并推广了相关领域的已有成果。
We construct a new class of biharmonic maps, which are the critical points for the bienergy functional, by deforming conformally the codomain metric of harmonic Riemannian submersions such that they become nonharmonic but biharmonic.
In this paper we present an explicit construction for the fundamental solution to the Dirac and Laplace operator on some non-orientable conformally flat manifolds. We first treat a class of projective cylinders and tori where we can study monogenic sections with values in different pin bundles. Then we discuss the Möbius strip, the Klein bottle and higher dimensional generalizations of them. We present integral representation formulas together with some elementary tools of harmonic analysis on these manifolds.
We introduce a functional that couples the nonlinear sigma model with a spinor field: $L=\int_M[|dϕ|^2+(ψ,\Dψ)]$. In two dimensions, it is conformally invariant. The critical points of this functional are called Dirac-harmonic maps. We study some geometric and analytic aspects of such maps, in particular a removable singularity theorem.
Conformal invariance plays a significant role in many areas of Physics, such as conformal field theory, renormalization theory, turbulence, general relativity. Naturally, it also plays an important role in geometry: theory of Riemannian surfaces, Weyl tensors, $Q$-curvature, Yang-Mills fields, etc... We shall be concerned with the study of conformal invariance in analysis. More precisely, we will focus on the study of nonlinear PDEs arising from conformally invariant two dimensional variational problems (e.g. harmonic maps, prescribed mean curvature surfaces, Willmore and Constrained conformal…
We construct a family of Hermitian metrics on the Hopf surface $ \mathbb{S}^3\times \mathbb{S}^1$, whose fundamental classes represent distinct cohomology classes in the Aeppli cohomology group. These metrics are locally conformally Kähler. Among the toric fibres of $π:\mathbb{S}^{3} \times \mathbb{S}^1\to\mathbb{C} P^1$ two of them are stable minimal surfaces and each of the two has a neighbourhood so that fibres therein are given by stable harmonic maps from 2-torus and outside, far away from the two tori, there are unstable harmonic ones that are also unstable minimal surfaces.
Motivated by the beautiful theory and the rich applications of harmonic conformal immersions and conformal immersions of constant mean curvature (CMC) surfaces, we study biharmonic conformal immersions of surfaces into a generic 3-manifold. We first derive an invariant equation for such immersions, we then try to answer the question, "what surfaces can be biharmonically conformally immersed into Euclidean 3-space?" We prove that a circular cylinder is the only CMC surface that can be biharmonically conformally immersed into a Euclidean 3-space; We obtain a classification of biharmonic conforma…
In this paper, we show several vanishing type theorems for $p$-harmonic $\ell$-forms on Riemannian manifolds ($p\geq2$). First of all, we consider complete non-compact immersed submanifolds $M^n$ of ${N}^{n+m}$ with flat normal bundle, we prove that any $p$-harmonic $\ell$-forms on $M$ is trivial if $N$ has pure curvature tensor and $M$ satisfies some geometric condition. Then, we obtain a vanishing theorem on Riemannian manifolds with weighted Poincaré inequality. Final, we investigate complete simply connected, locally conformally flat Riemannian manifolds $M$ and point out that there is no …
Let $(M^n, g)(n\geq3)$ be an $n$-dimensional complete Riemannian manifold with harmonic curvature and positive Yamabe constant. Denote by $R$ and $\mathring{Rm}$ the scalar curvature and the trace-free Riemannian curvature tensor of $M$, respectively. The main result of this paper states that $\mathring{Rm}$ goes to zero uniformly at infinity if for $p\geq \frac n2$, the $L^{p}$-norm of $\mathring{Rm}$ is finite. Moreover, If $R$ is positive, then $(M^n, g)$ is compact. As applications, we prove that $(M^n, g)$ is isometric to a spherical space form if for $p\geq \frac n2$, $R$ is positive and…
We show that on any Riemannian manifold with Hölder continuous metric tensor, there exists a $p$-harmonic coordinate system near any point. When $p = n$ this leads to a useful gauge condition for regularity results in conformal geometry. As applications, we show that any conformal mapping between manifolds having $C^α$ metric tensors is $C^{1+α}$ regular, and that a manifold with $W^{1,n} \cap C^α$ metric tensor and with vanishing Weyl tensor is locally conformally flat if $n \geq 4$. The results extend the works [LS14, LS15] from the case of $C^{1+α}$ metrics to the Hölder continuous case. In…
In this paper we study an analog of minimal surfaces called Weyl-minimal surfaces in conformal manifolds with a Weyl connection $(M^4,c,D)$. We show that there is an Eells-Salamon type correspondence between nonvertical $\mathcal{J}$-holomorphic curves in the weightless twistor space and branched Weyl-minimal surfaces. When $(M,c,J)$ is conformally almost-Hermitian, there is a canonical Weyl connection. We show that for the canonical Weyl connection, branched Weyl-minimal surfaces satisfy the adjunction inequality \begin{equation}\label{adj} χ(T_fΣ)+χ(N_fΣ) \le \pm c_1(f^*T^{(1,0)}M). \end{equ…
For a conformally compact Poincaré-Einstein manifold $(X,g_+)$, we consider two types of compactifications for it. One is $\bar{g}=ρ^2g_+$, where $ρ$ is a fixed smooth defining function; the other is the adapted (including Fefferman-Graham) compactification $\bar{g}_s=ρ^2_sg_+$ with a continuous parameter $s>\frac{n}{2}$. In this paper, we mainly prove that for a set of conformally compact Poincaré-Einstein manifolds $\{(X, g_{+}^{(i)})\}$ with conformal infinity of positive Yamabe type, $\{\bar{g}^{(i)}\}$ is compact in $C^{k,α}(\overline{X})$ topology if and only if $\{\bar{g}_s^{(i)}\}$ is …
Biharmonic Maps Between Conformally Compact Manifolds
研究共形紧致流形间双调和映射的非存在性，证明其在特定条件下为调和映射。
We succeed in writing 2-dimensional conformally invariant non-linear elliptic PDE (harmonic map equation, prescribed mean curvature equations...etc) in divergence form. This divergence free quantities generalize to target manifolds without symmetries the well known conservation laws for harmonic maps into homogeneous spaces. From this form we can recover, without the use of moving frame, all the classical regularity results known for 2-dimensional conformally invariant non-linear elliptic PDE . It enable us also to establish new results. In particular we solve a conjecture by E.Heinz asserting…
One of the main aims of this article is to give the complete classification of critical metrics of the volume functional on a compact manifold $M$ with boundary $\partial M$ and with harmonic Weyl tensor, which improves the corresponding classification for complete locally conformally flat case, due to Miao and Tam [18]. In particular, we prove that a critical metric with harmonic Weyl tensor on a simply connected compact manifold with boundary isometric to a standard sphere $\mathbb{S}^{n-1}$ must be isometric to a geodesic ball in a simply connected space form $\Bbb{R}^n,$ $\Bbb{H}^n$ and $\…
In this paper we study Moebius applicable surfaces, i.e., conformally immersed surfaces in Moebius 3-space which admit deformations preserving the Moebius metric. We show new characterizations of Willmore surfaces, Bonnet surfaces and Harmonic inverse mean curvature surfaces in terms of Moebius or similarity invariants.
In this paper we study Clifford and harmonic analysis on some conformal flat spin manifolds. In particular we treat manifolds that can be parametrized by $U / Γ$ where $U$ is a simply connected subdomain of either $S^{n}$ or $R^{n}$ and $Γ$ is a Kleinian group acting discontinuously on $U$. Examples of such manifolds treated here include for example $RP^{n}$ and $S^{1}\times S^{n-1}$. Special kinds of Clifford-analytic automorphic forms associated to the different choices of $Γ$ are used to construct Cauchy kernels, Cauchy Integral formulas, Green's kernels and formulas together with Hardy spa…
We study a functional, whose critical points couple Dirac-harmonic maps from surfaces with a two form. The critical points can be interpreted as coupling the prescribed mean curvature equation to spinor fields. On the other hand, this functional also arises as part of the supersymmetric sigma model in theoretical physics. In two dimensions it is conformally invariant. We call critical points of this functional magnetic Dirac-harmonic maps. We study geometric and analytic properties of magnetic Dirac-harmonic maps including their regularity and the removal of isolated singularities.
Willmore surfaces are the extremals of the Willmore functional (possibly under a constraint on the conformal structure). With the characterization of Willmore surfaces by the (possibly perturbed) harmonicity of the mean curvature sphere congruence [Blaschke, Ejiri, Rigoli, Burstall-Calderbank], a zero-curvature formulation follows [Burstall-Calderbank]. Deformations on the level of harmonic maps prove to give rise to deformations on the level of surfaces, with the definition of a spectral deformation [Burstall-Pedit-Pinkall, Burstall-Calderbank] and of a Baecklund transformation [Burstall-Quin…
Decreasing Weyl's energy by connected sums with locally conformally flat manifolds
研究两个四维流形连通和的Weyl函数，当其中一个为Bach平坦且另一个为局部共形平坦时，证明在特定条件下可构造更低Weyl能量的度量。
Maximally-warped metrics with harmonic curvature
本文研究了具有调和曲率的黎曼流形在允许最大数量局部扭曲积分解时的局部结构，并证明此类流形的局部等距类型构成有限维模空间，且存在非空开子集由非Ricci-平行且非共形平坦的局部不可约完备度量实现。
In this paper, by using monotonicity formulas for vector bundle-valued $p$-forms satisfying the conservation law, we first obtain general $L^2$ global rigidity theorems for locally conformally flat (LCF) manifolds with constant scalar curvature, under curvature pinching conditions. Secondly, we prove vanishing results for $L^2$ and some non-$L^2$ harmonic $p$-forms on LCF manifolds, by assuming that the underlying manifolds satisfy pointwise or integral curvature conditions. Moreover, by a Theorem of Li-Tam for harmonic functions, we show that the underlying manifold must have only one end. Fi…
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