Weiping Zhang
We establish an S^1-equivariant index theorem for Dirac operators on Z/k-manifolds. As an application, we generalize the Atiyah-Hirzebruch vanishing theorem for S^1-actions on closed spin manifolds to the case of Z/k-manifolds.
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Lower bound on the number of fixed points for circle actions on 10-dimensional almost complex manifolds
本文研究了10维近复流形上圆作用固定点数的下界，证明至少存在6个固定点，并给出了达到该下限的例子。
Circle actions on 6-dimensional oriented manifolds with 4 fixed points
本文分类了6维紧致定向流形上恰含4个不动点的圆周作用的不动点数据，证明其只能对应两个6球面旋转的不相交并集或 $\mathbb{CP}^3$ 上的线性作用，并涵盖 Petrie 奇异作用。
In this paper, we show that any compact manifold that carries a SL(n;R)-foliation is fibered on the circle S^1.
R-circles in general three dimensional CR manifolds (of contact type) are the analogues to traces of Lagrangian totally geodesic planes on the sphere viewed as the boundary of two dimensional complex hyperbolic space. They form a family of certain legendrian curves on the manifold. We prove that a diffeomorphism between three dimensional CR manifolds which preserve circles is either a CR diffeomorphism or conjugate CR diffeomorphism.
Let $Φ\colon \sbat \times M \to M$ be a smooth action of the unit circle $ \sbat$ on a manifold $M$. In this work, we compute the minimal model of $M$ in terms of the orbit space $B$ and the fixed point set $F\subset B$, as a dg-module over the Sullivan's minimal model of $B$.
Kawakubo and Uchida showed that, if a closed oriented $4k$-dimensional manifold $M$ admits a semi-free circle action such that the dimension of the fixed point set is less than $2k$, then the signature of $M$ vanishes. In this note, by using $G$-signature theorem and the rigidity of the signature operator, we generalize this result to more general circle actions. Combining the same idea with the remarkable Witten-Taubes-Bott rigidity theorem, we explore more vanishing results on spin manifolds admitting such circle actions. Our results are closely related to some earlier results of Conner-Floy…
We demonstrate an obstruction to finding certain splittings of four-manifolds along sufficiently twisted circle bundles over Riemann surfaces, arising from Seiberg-Witten theory. These obstructions are used to show a non-splitting result for algebraic surfaces of general type.
Circle actions on oriented manifolds with 3 fixed points
研究了具有3个不动点的紧致有向流形上的圆周作用，在各向同性子流形可定向的假设下，证明了8维情形下不动点权重与四元数射影空间作用一致，且不存在此类12维流形。
In this paper, we extend Gang Liu's three circle theorem for Kähler manifolds to almost Hermitian manifolds. As applications of the three circle theorem, we obtain sharp dimension estimates for spaces of holomorphic functions of polynomial growth and rigidity for the estimates, Liouville theorems for pluri-subharmonic functions of sub-logarithmic growth and Liouville theorems for holomorphic functions of Cheng-type, and fundamental theorem of algebra on almost Hermitian manifolds with slightly negative holomorphic sectional curvature. The results are generalization of Liu's results and some of…
This paper contains several results concerning circle action on almost-complex and smooth manifolds. More precisely, we show that, for an almost-complex manifold $M^{2mn}$(resp. a smooth manifold $N^{4mn}$), if there exists a partition $λ=(λ_{1},...,λ_{u})$ of weight $m$ such that the Chern number $(c_{λ_{1}}... c_{λ_{u}})^{n}[M]$ (resp. Pontrjagin number $(p_{λ_{1}}... p_{λ_{u}})^{n}[N]$) is nonzero, then \emph{any} circle action on $M^{2mn}$ (resp. $N^{4mn}$) has at least $n+1$ fixed points. When an even-dimensional smooth manifold $N^{2n}$ admits a semi-free action with isolated fixed point…
Consider a circle action on an 8-dimensional compact almost complex manifold with 4 fixed points. To the author's knowledge, $S^2 \times S^6$ is the only known example of such a manifold. In this paper, we prove that if the circle acts on an 8-dimensional compact almost complex manifold $M$ with 4 fixed points, all the Chern numbers and the Hirzebruch $χ_y$-genus of $M$ agree with those of $S^2 \times S^6$. In particular, $M$ is unitary cobordant to $S^2 \times S^6$.
We present some methods to construct smooth circle actions on symplectic manifolds with non-symplectic fixed point sets or non-symplectic cyclic isotropy point sets. All such actions are not compatible with any symplectic form.
We prove rigidity and vanishing theorems for several holomorphic Euler characteristics on complex contact manifolds admitting holomorphic circle actions preserving the contact structure. Such vanishings are reminiscent of those of LeBrun and Salamon on Fano contact manifolds but under a symmetry assumption instead of a curvature condition.
In this paper we show that the topological closure of the holonomy group of a certain class of projectively flat Finsler 2-manifolds of constant curvature is maximal, that is isomorphic to the connected component of the diffeomorphism group of the circle. This class of 2-manifolds contains the standard Funk plane of constant negative curvature and the Bryant-Shen-spheres of constant positive curvature. The result provides the first examples describing completely infinite dimensional Finslerian holonomy structures.
We study hyper-spheres, spheres and circles, with respect to an indefinite metric, in a tangent space on a 4-dimensional differentiable manifold. The manifold is equipped with a positive definite metric and an additional tensor structure of type (1, 1)11. The fourth power of the additional structure is minus identity and its components form a skew-circulant matrix in some local coordinate system. The both structures are compatible and they determine an associated indefinite metric on the manifold.
The aim of this paper is to study compact 5--manifolds which admit fixed point free circle actions. The first result implies that the torsion in the second homology and the second Stiefel--Whitney class have to satisfy strong restrictions. We then show that for simply connected 5--manifolds these restrictions are necessary and sufficient.
We prove that a compact 4-manifold which supports a circle-invariant fat SO(3)-bundle is diffeomorphic to either S^4 or CP^2-bar. The proof involves studying the resulting Hamiltonian circle action on an associated symplectic 6-manifold. Applying our result to the twistor bundle of Riemannian 4-manifolds shows that S^4 and CP^2-bar are the only 4-manifolds admitting circle-invariant metrics solving a certain curvature inequality. This can be seen as an analogue of Hsiang-Kliener's theorem that only S^4 and CP^2 admit circle-invariant metrics of positive sectional curvature.
The main results of this paper describes a formula for the Seiberg-Witten invariant of a 4-manifold which admits a nontrivial free S^1-action. We use this theorem to produce a nonsymplectic 4-manifold with a free circle action whose orbit space fibers over S^1. We also describe a 3-manifold which is not the orbit space of any symplectic 4-manifold with a free circle action. A corollary of the main theorem is a formula for the 3-dimensional Seiberg-Witten invariants of the total space of a circle bundle over a surface.
We give a lower bound for the degree of a finite cover of a hyperbolic 3-manifold which fibers over the circle, in terms of volume, the diameter of the manifold and other new invariants.
The author proved that if the circle acts symplectically on a compact, connected symplectic manifold $M$ with three fixed points, then $M$ is equivariantly symplectomorphic to some standard action on $\mathbb{CP}^2$. In this paper, we extend the result to a circle action on an almost complex manifold; if the circle acts on a compact, connected almost complex manifold $M$ with exactly three fixed points, then $\dim M=4$. Moreover, we deal with the cases of one fixed point and two fixed points.
We prove several versions of "quantization commutes with reduction" for circle actions on manifolds that are not symplectic. Instead, these manifolds possess a weaker structure, such as a spin^c structure. Our theorems work whenever the quantization data and the reduction data are compatible; this condition always holds if we start from a presymplectic (in particular, symplectic) manifold.
We consider a $3$-dimensional differentiable manifold with two circulant structures -- a Riemannian metric and an additional structure, whose third power is the identity. The structure is compatible with the metric such that an isometry is induced in any tangent space of the manifold. Further, we consider an associated metric with the Riemannian metric, which is necessary indefinite. We find equations of a sphere and of a circle, which are given in terms of the associated metric, with respect to the Riemannian metric.
We classify nonnegatively curved simply connected 4-manifolds with circle symmetry up to equivariant diffeomorphisms. The main problem is rule out knotted curves in the singular set of the orbit space. As an extension of this work we classify all knots in S^3 which can be realized as an extremal set with respect to an inner metric on S^3 which has nonnegative curvature in the Alexandrov sense.
We investigate geometric properties of indecomposable but non-irreducible Lorentzian manifolds, which are total spaces of circle bundles. We investigate under which conditions these manifolds are complete and give examples which fulfill the obtained conditions. In particular we investigate the Einstein equation for these spaces yielding examples for complete compact Ricci flat Lorentzian manifolds and manifolds with timelike Killing vector fields. Finally we study their holonomy and obtain in particular complete examples for Lorentzian manifolds with holonomy of so called type 4.
We exhibit a closed aspherical 5-manifold of nonpositive curvature that fibers over a circle whose fundamental group is hyperbolic relative to abelian subgroups such that the fiber is a closed aspherical 4-manifold whose fundamental group is not hyperbolic relative to abelian subgroups.
In this paper, we study a circle action on a compact oriented manifold with a discrete fixed point set. The fixed point data consists of the weights of the $S^1$-representations at the fixed points. We prove various results and properties of the action, in terms of the fixed point data. We show that the manifold can be described by a multigraph associated to it. Specializing into the case of dimension 4, we classify the fixed point data. Moreover, we prove that there exist oriented $S^1$-manifolds with these fixed point data. Finally, we show that a certain multigraph behaves like a manifold.
A construction is introduced for modifying hyperkaehler manifolds with tri-Hamiltonian circle action, that in favourable situations increases the second Betti number by one. This is based on the symplectic cut construction of Lerman. In 4 or 8 dimensions the construction may be interpreted as adding a D6-brane. A number of examples are given and generalisations to three-Sasaki and hypersymplectic geometry discussed.
Following the idea of Lusztig, Atiyah-Hirzebruch and Kosniowski, we note that the Dolbeault-type operators on compact, almost-complex manifolds are rigid. When the circle action has isolated fixed points, this rigidity result will produce many identities concerning the weights on the fixed points. In particular, it gives a criterion to detemine whether or not a symplectic circle action with isolated fixed points is Hamiltonian. As applications, we simplify the proofs of some known results related to symplectic circle actions, due to Godinho, Tolman-Weitsman and Pelayo-Tolman, and generalize so…
Circle actions on four dimensional almost complex manifolds with discrete fixed point sets
本文刻画紧近复4维流形上有效圆周作用离散不动点权重的充要条件，并据此否定陈数不等式猜想，推广至$\mathbb{T}^2$作用限制及复、辛与半自由情形。
The eta invariant of a circle bundle on a Fano manifold
研究正线丛上圆包的spin-c狄拉克算子，计算其eta不变量并扩展了adiabatic参数的范围。
We give examples of symplectic actions of a cyclic group, inducing a trivial action on homology, on four-manifolds that admit Hamiltonian circle actions, and show that they do not extend to Hamiltonian circle actions. Our work applies holomorphic methods to extend combinatorial tools developed for circle actions to study cyclic actions.
Circle actions on unitary manifolds with discrete fixed point sets
该文研究带离散不动点集的紧酉流形上的圆周作用，推广殆复流形相关结果，建立不动点权与多重图的关联，推导等变第一陈类结论，给出半自由作用下的Hirzebruch $χ_y$-亏格及不动点数下界。
This paper mainly focuses on the CR analogue of the three-circle theorem in a complete noncompact pseudohermitian manifold of vanishing torsion being odd dimensional counterpart of Kähler geometry. In this paper, we show that the CR three-circle theorem holds if its pseudohermitian sectional curvature is nonnegative. As an application, we confirm the first CR Yau's uniformization conjecture and obtain the CR analogue of the sharp dimension estimate for CR holomorphic functions of polynomial growth and its rigidity when the pseudohermitian sectional curvature is nonnegative. This is also the fi…
On a problem of Hopf for circle bundles over aspherical manifolds with hyperbolic fundamental groups
本文完全刻画了具有双曲基本群的闭可定向非球面流形上圆丛的绝对度大于一的自映射存在性，证明其等价于丛虚拟平凡，并由此验证了Hopf问题的强版本，同时构造了关于映射度单调的新数值不变量。
Circle actions on six dimensional oriented manifolds with isolated fixed points
本文研究了六维定向流形上圆作用的固定点数据，通过等变连通和操作将其简化为无固定点作用，揭示了固定点数据的可消去性。
In this paper we construct six-dimensional compact non-Kähler Hamiltonian circle manifolds which satisfy the strong Lefschetz property themselves but nevertheless have a non-Lefschetz symplectic quotient. This provides the first known counter examples to the question whether the strong Lefschetz property descends to the symplectic quotient. We also give examples of Hamiltonian strong Lefschetz circle manifolds which have a non-Lefschetz fixed point submanifold. In addition, we establish a sufficient and necessary condition for a finitely presentable group to be the fundamental group of a stron…
The goal of this paper is the construction of a compact manifold with G$_2$ holonomy and nodal singularities along circles using twisted connected sum method. This paper finds matching building blocks by solving the Calabi conjecture on certain asymptotically cylindrical manifolds with nodal singularities. However, by comparison to the untwisted connected sum case, it turns out that the obstruction space for the singular twisted connected sum construction is infinite dimensional. By analyzing the obstruction term, there are strong evidences that the obstruction may be resolved if a further glu…
Geometrization theorem, fibered case: Every three-manifold that fibers over the circle admits a geometric decomposition. Double limit theorem: for any sequence of quasi-Fuchsian groups whose controlling pair of conformal structures tends toward a pair of projectively measured laminations that bind the surface, there is a convergent subsequence. This preprint also analyzes the quasi-isometric geometry of quasi-Fuchsian 3-manifolds. This eprint is based on a 1986 preprint, which was refereed and accepted for publication, but which I neglected to correct and return. The referee's corrections have…
We give topological obstructions to the existence of a closed exact Lagrangian submanifold in the cotangent bundle of a closed manifold M which is the total space of a fibration over the circle. For instance we show that the fundamental group of such a Lagrangian submanifold cannot be the free product of two non-trivial groups and that in any finite presentation of this group the difference between the number of generators and the number of relations is less than two.
We show that a complete simply-connected hyperkaehler 4-manifold with an isometric triholomorphic circle action is obtained from the Gibbons-Hawking ansatz with some suitable harmonic function.
A Complete Derivation of Complex Circle Manifold (CCM) Riemannian manifold Optimization Equations
本文针对复圆流形（CCM）在MIMO等应用中缺乏严谨推导的问题，系统证明了其切空间和黎曼梯度等关键性质，为CCM流形优化提供了统一的权威参考。
Let $M$ be a symplectic manifold, equipped with a semifree symplectic circle action with a finite, nonempty fixed point set. We show that the circle action must be Hamiltonian, and $M$ must have the equivariant cohomology and Chern classes of $(P^1)^n$.
We investigate the maximal solid tubes around short simple geodesics in hyperbolic three-manifolds and how complex length of curves relate to closed, incompressible, least area minimal surfaces. As applications, we prove, there are some closed hyperbolic three-manifolds fibering over the circle which are not foliated by closed incompressible minimal surfaces diffeomorphic to the fiber. We also show, the existence of quasi-Fuchsian manifolds containing arbitrarily many embedded closed incompressible minimal surfaces. Our strategy is to prove main theorems under natural geometric conditions on t…
Topological and dynamical aspects of some spectral invariants of contact manifolds with circle action
该研究针对带横向圆周作用与酉表示的任意维CR切触流形，分析其解析挠率与类eta不变量，证明相关谱级数可同时从拓扑角度和Reeb流的纯动力学角度整体解读。
Free Circle Actions and Positive Ricci Curvature on Manifolds with the Cohomology Ring of $S^2\times S^5$
研究具有 $S^2\times S^5$ 上同调环的闭连通的spin 7流形中，672种定向微分类型中哪些允许自由圆作用，并展示当存在此类作用时，存在无限多个非等价的自由圆作用，且在大多数情况下构造正曲率不变黎曼度量。
Given a smooth, closed, oriented 4-manifold X and alpha in H_2(X,Z) such that alpha.alpha > 0, a closed 2-form w is constructed, Poincare dual to alpha, which is symplectic on the complement of a finite set of unknotted circles. The number of circles, counted with sign, is given by d = (c_1(s)^2 -3sigma(X) -2chi(X))/4, where s is a certain spin^C structure naturally associated to w.
Circle bundles with PSC over large manifolds
本研究利用辛几何技术构造了无穷多类m≥4维宏观大体积流形的正标量曲率圆丛实例，其全空间宏观维度不超过⌈m/2⌉+1，并全维度解答了Gromov关于可扩流形上此类圆丛存在性的问题。
Let $X$ be a compact strongly pseudoconvex CR manifold with a transversal CR $S^1$-action. In this paper, we establish the asymptotic expansion of Szegő kernels of positive Fourier components and by using the asymptotics, we show that $X$ can be equivariant CR embedded into some $\mathbb C^N$ equipped with a simple $S^1$-action. An equivariant embedding of quasi-regular Sasakian manifold is also derived.
The rigidity of conformal circle-preserving transformations on Berwaldian manifolds
本文证明，若完备Berwald流形存在非平凡保圆共形变换，且其旗曲率非零的点稠密（如正或负旗曲率），则该流形必为黎曼流形。
We determine the Seiberg-Witten-Floer homology groups of the three-manifold which is the product of a surface of genus $g \geq 1$ times the circle, together with its ring structure, for spin-c structures which are non-trivial on the three-manifold. We give applications to computing Seiberg-Witten invariants of four-manifolds which are connected sums along surfaces and also we reprove the higher type adjunction inequalities previously obtained by Oszváth and Szabó.
Motivated by the moduli theory of taut contact circles on spherical 3-manifolds, we relate taut contact circles to transversely holomorphic flows. We give an elementary survey of such 1-dimensional foliations from a topological viewpoint. We describe a complex analogue of the classical Godbillon-Vey invariant, the so-called Bott invariant, and a logarithmic monodromy of closed leaves. The Bott invariant allows us to formulate a generalised Gauß-Bonnet theorem. We compute these invariants for the Poincaré foliations on the 3-sphere and derive rigidity statements, including a uniformisation theo…
We obtain a topological and weakly equivariant classification of closed three-dimensional Alexandrov spaces with an effective isometric circle action. As an application of the classification we prove a version of the Borel conjecture for closed three-dimensional Alexandrov spaces with circle symmetry.
Hyperbolic 4-manifolds with perfect circle-valued Morse functions
本文构造了具有完美圆值Morse函数的有限体积双曲4-流形，证明了存在无限多个此类流形，其具有有界数量的1-和3-把手分解及有界的Betti数。
Equivariant Kodaira embedding of CR manifolds with circle action
本文研究带圆周作用的紧CR流形，通过证明高张量幂下加权Fourier-Szegő核的完全渐近展开，建立了等变Kodaira嵌入定理。
In this paper, it is shown that non-isomorphic effective linear circle actions yield non-diffeomorphic differential structures on the corresponding orbit spaces.
The main result of this paper is a formula for calculating the Seiberg-Witten invariants of 4-manifolds with fixed-point free circle actions. This is done by showing under suitable conditions the existence of a diffeomorphism between the moduli space of the 4-manifold and the moduli space of the quotient 3-orbifold. Two corollaries include b_+>1 4-manifolds with fixed-point free circle actions are simple type and a new proof that the four dimensional invariants of $Y \times S^1$ are equal to the the three dimensional invariants of $Y$. An infinite number of b_+=1 4-manifolds where the Seiberg-…
The classical Hadamard three circle theorem is generalized to complete Kähler manifolds. More precisely, we show that the nonnegativity of the holomorphic sectional curvature is a necessary and sufficient condition for the three circle theorem. As corollaries, two sharp monotonicity formulae for holomorphic functions are derived. Among applications, we derive sharp dimension estimates (with rigidity) of holomorphic functions with polynomial growth when the holomorphic sectional curvature is nonnegative. When the bisectional curvature is nonnegative, this was due to Ni. Also we study holomorphi…
We obtain a topological and equivariant classification of closed, connected three-dimensional Alexandrov spaces admitting a local isometric circle action. We show, in particular, that such spaces are homeomorphic to connected sums of some closed 3-manifold with a local circle action and finitely many copies of the suspension of the real projective plane.
Let X be a compact oriented Riemannian manifold and let $ϕ:X\to S^1$ be a circle-valued Morse function. Under some mild assumptions on $ϕ$, we prove a formula relating: (a) the number of closed orbits of the gradient flow of $ϕ$ of any given degree; (b) the torsion of a ``Morse complex'', which counts gradient flow lines between critical points of $ϕ$; and (c) a kind of Reidemeister torsion of X determined by the homotopy class of $ϕ$. When $\dim(X)=3$ and $b_1(X)>0$, we state a conjecture analogous to Taubes's ``SW=Gromov'' theorem, and we use it to deduce (for closed manifolds, modulo signs)…
Wave maps from circle to Riemannian manifold: global controllability is equivalent to homotopy
研究圆到紧Riemannian流形的波映射，证明全局能控性由数据同伦类精确刻画，建立稳态间一致时间全局能控性，并得出负截面曲率闭测地线附近的定量指数稳定性。
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