Christian Lange
We characterize Riemannian orbifolds and their coverings in terms of metric geometry. In particular, we show that the metric double of a Riemannian orbifold along the closure of its codimension one stratum is a Riemannian orbifold and that the natural projection is an orbifold covering.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In a compact orbifold, for small prescribed volume, an isoperimetric region is close to a small metric ball; in a Euclidean orbifold, it is a small metric ball.
This work proves certain general orbifold compactness results for spaces of Riemannian metrics, generalizing earlier results along these lines for Einstein metrics or metrics with bounded Ricci curvature. This is then applied to prove such compactness for spaces of Bach-flat (for example half-conformally flat) metrics on 4-manifolds, and related results for metrics which are critical points of other natural Riemannian functionals on the space of metrics.
A construction of Kaehler-Einstein metrics using Galois coverings, studied by Arezzo-Ghigi-Pirola, is generalized to orbifolds. By applying it to certain orbifold covers of P^n which are trivial set theoretically, one obtains new Einstein metrics on odd-dimensional spheres. The method also gives Kaehler-Einstein metrics on degree 2 Del Pezzo surfaces with A_1 or A_2-singularities.
Orbifolds and orbibundles in complex hyperbolic geometry
本文利用diffeology从几何角度发展orbibundle理论，提供新工具以计算4-流形几何中底空间为2-orbifold的复双曲圆盘orbibundle的欧拉数与PU(2,1)-表示的Toledo不变量。
For a holomorphic vector bundle over a compact Kähler orbifold, the slope stability of the bundle is shown to be equivalent to the existence of a Hermitian-Einstein metric or to the properness of a certain functional introduced by Donaldson.
In this note, we prove that any non-collapsing and compact Gromov-Hausdorff limit of Kahler-Einstein manifolds is either smooth or is orbifold outside a subvariety of complex codimension at least 3.
A general collapsing result for families of stratified Riemannian metrics on orbifolds
本文提出了一种在轨型流形上一族轨型黎曼度量的通用坍缩结果，并引入了新的轨型几何结构以支持该定理的证明。
A note on orbifold regularity of canonical metrics
本文证明紧致Kähler簇（对数终端奇点且第一陈类为零）上任意奇异Ricci平坦Kähler度量在轨道折叠轨迹上具有轨道折叠奇点。
In this paper we compute the Futaki invariant of adiabatic Kaehler classes on resolutions of Kaehler orbifolds with isolated singularities. Combined with previous existence results of extremal metrics by Arezzo-Lena-Mazzieri, this gives a number of new existence and non-existence results for cscK metrics.
In this short note, we investigate the existence of orbifold Kähler-Einstein metrics on toric varieties. In particular, we show that every $\mathbb{Q}$-factorial normal projective toric variety allows an orbifold Kähler-Einstein metric. Moreover, we characterize the K-stability of $\mathbb{Q}$-factorial toric pairs of Picard number one in terms of the log Cox ring and the universal orbifold cover.
We embed polarised orbifolds with cyclic stabiliser groups into weighted projective space via a weighted form of Kodaira embedding. Dividing by the (non-reductive) automorphisms of weighted projective space then formally gives a moduli space of orbifolds. We show how to express this as a reductive quotient and so a GIT problem, thus defining a notion of stability for orbifolds. We then prove an orbifold version of Donaldson's theorem: the existence of an orbifold Kahler metric of constant scalar curvature implies K-semistability. By extending the notion of slope stability to orbifolds we there…
Recently, Atiyah and LeBrun proved versions of the Gauss-Bonnet and Hirzebruch signature Theorems for metrics with edge-cone singularities in dimension four, which they applied to obtain an inequality of Hitchin-Thorpe type for Einstein edge-cone metrics. Interestingly, many natural examples of edge-cone metrics in dimension four are anti-self-dual (or self-dual depending upon choice of orientation). On such a space there is an important elliptic complex called the anti-self-dual deformation complex, whose index gives crucial information about the local structure of the moduli space of anti-se…
Consider an Einstein orbifold $(M_0,g_0)$ of real dimension $2n$ having a singularity with orbifold group the cyclic group of order $n$ in ${\rm{SU}}(n)$ which is generated by an $n$th root of unity times the identity. Existence of a Ricci-flat Kähler ALE metric with this group at infinity was shown by Calabi. There is a natural "approximate" Einstein metric on the desingularization of $M_0$ obtained by replacing a small neighborhood of the singular point of the orbifold with a scaled and truncated Calabi metric. In this paper, we identify the first obstruction to perturbing this approximate E…
Constant Scalar Curvature Kähler Metrics on Resolutions of an Orbifold Singularity of Depth 1
本文针对沿高复余维子轨形具有I型奇点的常标量曲率Kähler轨形，构造其适当解消上的常标量曲率Kähler度量新例。
An orbifold version of the Hitchin-Thorpe inequality is used to prove that certain weighted projective spaces do not admit orbifold Einstein metrics. Also, several estimates for the orbifold Yamabe invariants of weighted projective spaces are proved.
Diffeological Riemannian orbifolds
研究可微栈上黎曼度量与微分几何轨道空间上黎曼度量数据的等价性，使用特定框架，证明经典黎曼orbifold与黎曼微分几何orbifold概念等价，还探讨李群胚上黎曼度量诱导条件及恰当性条件。
Moduli spaces of flat Riemannian metrics on orbifolds
本文研究闭黎曼轨形的平坦度量模空间，证明其 Teichmüller 空间可分类映射类群的真作用，且模空间本身是非常好的轨形，并在技术假设下给出其到可公度性的代数描述。
We present an analytic construction of complete non-compact 8-dimensional Ricci-flat manifolds with holonomy Spin(7). The construction relies on the study of the adiabatic limit of metrics with holonomy Spin(7) on principal Seifert circle bundles over asymptotically conical G2 orbifolds. The metrics we produce have an asymptotic geometry, so-called ALC geometry, that generalises to higher dimensions the geometry of 4-dimensional ALF hyperkähler metrics. We apply our construction to asymptotically conical G2 metrics arising from self-dual Einstein 4-orbifolds with positive scalar curvature. As …
We prove that any compact selfdual Einstein 4-orbifold of positive scalar curvature whose isometry group contains a 2-torus is, up to an orbifold covering, a quaternion Kaehler quotient of (k-1)-dimensional quaternionic projective space by a (k-2)-torus for some $k\geq 2$. We also obtain a topological classification in terms of the intersection form of the 4-orbifold.
We consider the self-dual conformal classes on n#CP^2 discovered by LeBrun. These depend upon a choice of n points in hyperbolic 3-space, called monopole points. We investigate the limiting behavior of various constant scalar curvature metrics in these conformal classes as the points approach each other, or as the points tend to the boundary of hyperbolic space. There is a close connection to the orbifold Yamabe problem, which we show is not always solvable (in contrast to the case of compact manifolds). In particular, we show that there is no constant scalar curvature orbifold metric in the c…
In this paper we give sufficient conditions on a compact orbifold with an extremal Kaehler metric to admit a resolution with an extremal Kaehler metric. We also complete the Kaehler constant scalar curvature case.
A theorem of E.Lerman and S.Tolman, generalizing a result of T.Delzant, states that compact symplectic toric orbifolds are classified by their moment polytopes, together with a positive integer label attached to each of their facets. In this paper we use this result, and the existence of "global" action-angle coordinates, to give an effective parametrization of all compatible toric complex structures on a compact symplectic toric orbifold, by means of smooth functions on the corresponding moment polytope. This is equivalent to parametrizing all toric Kahler metrics and generalizes an analogous…
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-…
We show that suitably defined systolic ratios are globally bounded from above on the space of rotationally symmetric spindle orbifolds and that the upper bound is attained precisely at so-called Besse metrics, i.e. Riemannian orbifold metrics all of whose geodesics are closed.
We show that the geodesic period spectrum of a Riemannian 2-orbifold all of whose geodesics are closed depends, up to a constant, only on its orbifold topology and compute it. In the manifold case we recover the fact proved by Gromoll, Grove and Pries that all prime geodesics have the same length. In the appendix we partly strengthen our result in terms of conjugacy of contact forms and explain how to deduce rigidity on the real projective plane based on a systolic inequality due to Pu. (We do not use a Lusternik-Schnirelmann type theorem on the existence of at least three simple closed geodes…
We show that the moduli space of nonnegatively curved metrics on each member of a large class of 2-connected 7-manifolds, including each smooth manifold homeomorphic to $S^7$, has infinitely many connected components. The components are distinguished using the Kreck-Stolz $s$-invariant computed for metrics constructed by Goette, Kerin and Shankar. The invariant is computed by extending each metric to the total space of an orbifold disc bundle and applying generalizations of the Atiyah-Patodi-Singer index theorem for orbifolds with boundary. We develop methods for computing characteristic class…
An index theorem for the anti-self-dual deformation complex on anti-self-dual orbifolds with singularities conjugate to ADE-type is proved. In 1988, Claude Lebrun gave examples of scalar-flat Kähler ALE metrics with negative mass, on the total space of the bundle $\mathcal{O}(-n)$ over $S^2$. A corollary of this index theorem is that the moduli space of anti-self-dual ALE metrics near each of these metrics has dimension at least $4n-12$, and thus for $n \geq 4$ the LeBrun metrics admit a plethora of non-trivial anti-self-dual deformations.
We construct a Laplace isospectral deformation of metrics on an orbifold quotient of a nilmanifold. Each orbifold in the deformation contains singular points with order two isotropy. Isospectrality is obtained by modifying a generalization of Sunada's Theorem due to DeTurck and Gordon.
We give an explicit local classification of conformally equivalent but oppositely oriented Kaehler metrics on a 4-manifold which are toric with respect to a common 2-torus action. In the generic case, these structures have an intriguing local geometry depending on a quadratic polynomial and two arbitrary functions of one variable, these two functions being explicit degree 4 polynomials when the Kaehler metrics are extremal (in the sense of Calabi). One motivation for and application of this result is an explicit local description of Einstein 4-manifolds which are hermitian with respect to eith…
Geodesics on a K3 Surface near the Orbifold Limit
本文研究接近轨形极限的Kummer K3曲面，改进Calabi-Yau度量估计，并利用这些估计分析稳定闭测地线的存在性限制及高对称情形下的存在性。
In this paper we consider the Boothby-Wang construction over twist 1 stage 3 Bott orbifolds given in terms of the log pair $(S_{\bf n},Δ_{\bf m})$. We give explicit constant scalar curvature (CSC) Sasaki metrics either directly from CSC Kähler orbifold metrics or by using the weighted extremal approach of Apostolov and Calderbank. The Sasaki 7-manifolds (orbifolds) are finitely covered by compact simply connected manifolds (orbifolds) with the rational homology of the 2-fold connected sum of $S^2\times S^5$.
A theorem of Anderson and Bando-Kasue-Nakajima from 1989 states that to compactify the set of normalized Einstein metrics with a lower bound on the volume and an upper bound on the diameter in the Gromov-Hausdorff sense, one has to add singular spaces called Einstein orbifolds, and the singularities form as blow-downs of Ricci-flat ALE spaces. This raises some natural issues, in particular: can all Einstein orbifolds be Gromov-Hausdorff limits of smooth Einstein manifolds? Can we describe more precisely the smooth Einstein metrics close to a given singular one? In this first paper, we prove th…
A proof of Kobayashi's properness criterion from a viewpoint of metric geometry
本文将 Kobayashi 关于实约化 Lie 群齐性空间上真作用的 KAK 判据推广到可在度量空间上等距作用的局部紧群，并借助非紧对称空间的 CAT(0) 几何重新证明该判据。
On the existence of conic Sasaki-Einstein metrics on log Fano Sasakian manifolds of dimension five
本文研究五维紧对数法诺Sasaki流形上的锥形Sasaki-Ricci流，得到曲率一致界，证明流的解收敛到奇异轨形孤立子，并在横向对数K-多稳定条件下得到锥形Sasaki-Einstein度量的存在性。
We provide an explicit resolution of the existence problem for extremal Kaehler metrics on toric 4-orbifolds M with second Betti number b2(M)=2. More precisely we show that M admits such a metric if and only if its rational Delzant polytope (which is a labelled quadrilateral) is K-polystable in the relative, toric sense (as studied by S. Donaldson, E. Legendre, G. Szekelyhidi et al.). Furthermore, in this case, the extremal Kaehler metric is ambitoric, i.e., compatible with a conformally equivalent, oppositely oriented toric Kaehler metric, which turns out to be extremal as well. These results…
The harmonic $2$-forms on $K3$ surfaces converging to a flat $4$-dimensional orbifold
研究K3曲面上 Ricci-flat Kähler度量下调和2形式的渐进行为，发现反自对偶调和2形式空间分解为两个子空间，一个收敛到 торus 商空间的平坦2形式，另一个收敛到ALE空间上的反自对偶连接的第一陈形式。
This is a survey on the correspondence between asymptotically complex hyperbolic Einstein metrics and CR structures on the boundary at infinity, which is the complex version of that between Poincaré-Einstein metrics and conformal structures. We mainly discuss existence theorems, and propose several open problems.
We find new obstructions to the desingularization of compact Einstein orbifolds by smooth Einstein metrics. These new obstructions, specific to the compact situation, raise the question of whether a compact Einstein $4$-orbifold which is limit of Einstein metrics bubbling out Eguchi-Hanson metrics has to be Kähler. We then test these obstructions to discuss if it is possible to produce a Ricci-flat but not Kähler metric by the most promising desingularization configuration proposed by Page in 1981. We identify $84$ obstructions which, once compared to the $57$ degrees of freedom, indicate that…
The energy of maps accompanying the collapsing of the $K3$ surface to a flat 3-dimensional orbifold
本文研究 $K3$ 曲面超 Kähler 度量塌缩到平坦 3 维轨形体时的映射能量，引入同伦类不变量作为能量下界，并证明 Foscolo 塌缩族中能量与不变量之比收敛到 1。
The aim of this paper is to construct infinitely many families of Einstein metrics on the connected sums of arbitrary number of copies of $S^2\times S^3$. We realize these 5-manifolds as total spaces of Seifert bundles over Del Pezzo orbifolds. A Kähler--Einstein metric on the Del Pezzo orbifold is then lifted to an Einstein metric using the Kobayashi--Boyer--Galicki method.
We present explicit constructions of complete Ricci-flat Kahler metrics that are asymptotic to cones over non-regular Sasaki-Einstein manifolds. The metrics are constructed from a complete Kahler-Einstein manifold (V,g_V) of positive Ricci curvature and admit a Hamiltonian two-form of order two. We obtain Ricci-flat Kahler metrics on the total spaces of (i) holomorphic C^2/Z_p orbifold fibrations over V, (ii) holomorphic orbifold fibrations over weighted projective spaces WCP^1, with generic fibres being the canonical complex cone over V, and (iii) the canonical orbifold line bundle over a fam…
Let O be a symplectic toric 2n-dimensional orbifold with a fixed T^n-action and with a toric Kahler metric g. We previously explored whether, when O is a manifold, the equivariant spectrum of the Laplace operator acting on smooth functions on (O,g) determines the moment polytope of O, and hence by Delzant's theorem determines O up to symplectomorphism. In the setting of toric orbifolds we significantly improve upon our previous results and show that the moment polytope of a generic toric orbifold is determined by its equivariant spectrum, up to two possibilities and up to translation. This inv…
We study the geometry and topology of Riemannian 3-orbifolds which are locally volume collapsed with respect to a curvature scale. We show that a sufficiently collapsed closed 3-orbifold without bad 2-suborbifolds either admits a metric of nonnegative sectional curvature or satisfies Thurston's Geometrization Conjecture. We also prove a version with boundary. Kleiner and Lott independently proved similar results.
Seiberg-Witten theory is used to obtain new obstructions to the existence of Einstein metrics on 4-manifolds with conical singularities along an embedded surface. In the present article, the cone angle is required to be of the form 2(pi)/p, where p is a positive integer, but we conjecture that similar results will also hold in greater generality.
Moduli Space of Bi-Invariant Metrics
本研究针对李群中双不变度量的分类问题，证明双不变度量模空间$\frak{BI}$为轨形，并给出$\frak{BI}$与等价双不变度量空间$\frak{EBI}$的显式刻画，推进了李群几何的相关研究。
The Kummer Construction of Calabi-Yau and Hyper-Kähler Metrics on the $K3$ Surface, and Large Families of Volume Non-collapsed Limiting Compact Hyper-Kähler Orbifolds
本文基于奇异摄动与加权函数空间分析，给出佩奇等关于K3曲面里奇平坦凯勒度量库默构造猜想的两个新严格证明，得到改进估计、完整58维模空间及体积非塌缩极限轨形大家族。
We show that if $g$ is a Riemannian metric on a closed piecewise locally symmetric manifold $M$, then the lift of $g$ to the universal cover $\widetilde{M}$ has a discrete isometry group. We also show that the index $[\Isom(\widetilde{M}): π_1(M)]$ is bounded by a constant independent of $g$.
Gromov proposed to extract the (differential) geometric content of a sub-riemannian space exclusively from its Carnot-Carathéodory distance. One of the most striking features of a regular sub-riemannian space is that it has at any point a metric tangent space with the algebraic structure of a Carnot group, hence a homogeneous Lie group. Siebert characterizes homogeneous Lie groups as locally compact groups admitting a contracting and continuous one-parameter group of automorphisms. Siebert result has not a metric character. In these notes I show that sub-riemannian geometry may be described by…
Metric Lie Groups. Carnot-Carathéodory spaces from the homogeneous viewpoint
本书探讨 Lie 群上左不变距离定义的非光滑几何，重点阐述 Carnot 群等度量 Lie 群在幂零群渐近锥、对称空间抛物边界及流形极限等数学语境中的核心作用。
We find a new obstruction for a real Einstein 4-orbifold with an A1-singularity to be a limit of smooth Einstein 4-manifolds. The obstruction is a curvature condition at the singular point. For asymptotically hyperbolic metrics, with boundary at infinity a conformal metric, we prove that if the obstruction vanishes, one can desingularize Einstein orbifolds with such singularities. The Dirichlet problem consists in finding Einstein metrics with given conformal infinity on the boundary: we prove that our obstruction defines a wall in the space of conformal metrics on the boundary, and that all t…
We show that on smooth minimal surfaces of general type, the Kähler-Ricci flow starting at any initial Kähler metric converges in the Gromov-Hausdorff sense to a Kähler-Einstein orbifold surface. In particular, the diameter of the evolving metrics is uniformly bounded for all time and the Kähler-Ricci flow contracts all the holomorphic spheres with $(-2)$ self-intersection number to isolated orbifold points. Our estimates do not require a priori the existence of an orbifold Kähler-Einstein metric on the canonical model.
We prove the existence of an abundance of new Einstein metrics on odd dimensional spheres including exotic spheres, many of them depending on continuous parameters. The number of families as well as the number of parameter grows double exponentially with the dimension. Our method of proof uses Brieskorn-Pham singularities to realize spheres (and exotic spheres) as circle orbi-bundles over complex algebraic orbifolds, and lift a Kaehler-Einstein metric from the orbifold to a Sasakian-Einstein metric on the sphere.
In this paper we prove that generic small partial smoothings of Kahler-Einstein (KE) Del Pezzo orbifolds with only nodal singularities, and with no non-zero holomorphic vector fields, admit orbifold KE metrics which are close in the Gromov-Hausdorff sense to the original KE metric.
We show that any compact convex simple lattice polytope is the moment polytope of a Kähler-Einstein orbifold, unique up to orbifold covering and homothety. We extend the Wang-Zhu Theorem \cite{WZ} giving the existence of a Kähler-Ricci soliton on any toric monotone manifold on any compact convex simple labelled polytope satisfying the combinatoric condition corresponding to monotonicity. We obtain that any compact convex simple polytope $P\subset \bR^n$ admits a set of inward normals, unique up to dilatation, such that there exists a symplectic potential satisfying the Guillemin boundary condi…
We provide a new angle and obtain new results on a class of metrics on length-normalized curves in $d$ dimensions, represented by their unit tangents expressed as a function of arc-length, which are functions from the unit interval to the $(d-1)$-dimensional unit sphere. These metrics are derived from the combined action of diffeomorphisms (change of parameters) and arc-length-dependent rotation acting on the tangent. Minimizing a Riemannian metric balancing a right-invariant metric on diffeomorphisms and an $L^2$ norm on the motion of tangents leads to a special case of "metamorphosis", which…
We construct complete noncompact Riemannian metrics with $G_2$-holonomy on noncompact orbifolds that are $\Bbb R^3$-bundles with the twistor space $\mathcal{Z}$ as a spherical fiber.
On statistics which are almost sufficient from the viewpoint of the Fisher metrics
本文提出Fisher度量下几乎充分的统计量概念，要求诱导模型与原模型度量双Lipschitz等价，并利用条件概率或密度分解给出类似充分统计量的刻画。
In this paper we study the sub-Finsler geometry as a time-optimal control problem. In particular, we consider non-smooth and non-strictly convex sub-Finsler structures associated with the Heisenberg, Grushin, and Martinet distributions. Motivated by problems in geometric group theory, we characterize extremal curves, discuss their optimality, and calculate the metric spheres, proving their Euclidean rectifiability.
Given a (possibly singular) Riemannian foliation $\mathcal{F}$ with closed leaves on a compact manifold $M$ with an adapted metric, we investigate the wave trace invariants for the basic Laplacian about a non-zero period. We compare them to the wave invariants of the underlying Riemannian orbifold that exists when the leaves in the regular region are identified to points, equipped with the metric that is transverse to the leaves of the foliation. Recalling that the basic Laplacian differs from the underlying orbifold Laplacian by a term that is the mean curvature vector field associated to the…
Blowing up Chern-Ricci flat balanced metrics
本文将 Arezzo-Pacard 构造推广到平衡情形，证明陈-里奇平坦平衡轨形在光滑点爆破上存在常陈数量曲率平衡度量，并证明具有 crepant 消解的孤立奇点轨形总带有陈-里奇平坦平衡度量。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。