Foliations invariant under Lie group transverse actions

Compact Lie algebras, transversely Lie foliations and fibrations

We study Lie foliations on compact manifolds, in case the Lie group is compact. Our main results improve Tischler classical result on the existence of fibration and, as an application, we study the case the manifold has an amenable fundamental group.

Rigidity of pseudo-Anosov flows transverse to R-covered foliations

A foliation is R-covered if the leaf space in the universal cover is homeomorphic to the real numbers. We show that, up to topological conjugacy, there are at most two pseudo-Anosov flows transverse to such a foliation. If there are two, then the foliation is weakly conjugate to the the stable foliation of an R-covered Anosov flow. The proof uses the universal circle to R-covered foliations.

横截纤维纽结与紧叶状结构

Transverse fibered knots and Taut Foliations

本研究给出Honda-Kazec-Matic定理的简短替代证明，验证了满足特定条件的纤维纽结所支撑的切触结构可由紧叶状结构微扰得到，为相关几何拓扑结论提供了新的证明路径。

转向三角剖分与横截叶状结构

Veering triangulations and transverse foliations

本文提出组合方法研究伪Anosov流的横截叶状结构存在性，证明余维1横截叶状结构由转向三角剖分的分支曲面承载，结合Massoni工作将问题转化为Homeo₊([0,1])上不等式组可行性，并以纽结10₁₄₅的Dehn手术为例验证结论。

Monopoles and foliations without holonomy-invariant transverse measure

This article proves a uniform exponential decay estimate for Seiberg-Witten equations on non-compact 4-manifolds with exact symplectic ends of bounded geometry. This is an extension of the analysis for asymptotically flat almost Kähler (AFAK) structures by Kronheimer and Mrowka. As an application, we construct an invariant for smooth foliations without holonomy-invariant transverse measure, which takes value in the boundary-stable version of the monopole Floer homology group, without invoking the Eliashberg-Thurston perturbation.

S-stable foliations on flow-spines with transverse Reeb flow

The notion of S-stability of foliations on branched simple polyhedrons is introduced by R. Benedetti and C. Petronio in the study of characteristic foliations of contact structures on 3-manifolds. We additionally assume that the 1-form $β$ defining a foliation on a branched simple polyhedron $P$ satisfies $dβ>0$, which means that the foliation is a characteristic foliation of a contact form whose Reeb flow is transverse to $P$. In this paper, we show that if there exists a 1-form $β$ on $P$ with $dβ>0$ then we can find a 1-form with the same property and additionally being S-stable. We then pr…

Foliations Transverse to Triangulations of 3-Manifolds

We investigate the combinatorial analogues, in the context of normal surfaces, of taut and transversely measured (codimension 1) foliations of 3-manifolds. We establish that the existence of certain combinatorial structures, a priori weaker than the existence of the corresponding foliation, is sufficient to guarantee that the manifold in question satisfies certain properties, e.g. irreducibility. The finiteness of our combinatorial structures allows us to make our results quantitative in nature and has (coarse) geometrical consequences for the manifold. Furthermore, our techniques give a strai…

Geometry of foliations and flows I: Almost transverse pseudo-Anosov flows and asymptotic behavior of foliations

Let F be a foliation in a closed 3-manifold with negatively curved fundamental group and suppose that F is almost transverse to a quasigeodesic pseudo-Anosov flow. We show that the leaves of the foliation in the universal cover extend continuously to the sphere at infinity, hence the limit sets are continuous images of the circle. One important corollary is that if F is a Reebless finite depth foliation in a hyperbolic manifold, then it has the continuous extension property. Such finite depth foliations exist whenever the second Betti number is non zero. The result also applies to other classe…

Transverse noncommutative geometry of foliations

We define an $L^2$-signature for proper actions on spaces of leaves of transversely oriented foliations with bounded geometry. This is achieved by using the Connes fibration to reduce the problem to the case of Riemannian bifoliations where we show that any transversely elliptic first order operator in an appropriate Beals-Greiner calculus, satisfying the usual axioms, gives rise to a semi-finite spectral triple over the crossed product algebra of the foliation by the action, and hence a periodic cyclic cohomology class through the Connes-Chern character. The Connes-Moscovici hypoelliptic sign…

球面丛中的仿射横截叶状结构

Affine transverse foliations in sphere bundles

本文研究定向球面丛上的仿射横截叶状结构，给出了丛的欧拉数的上界，并利用初等方法证明了当底流形的基本群是可解群时欧拉数为零。

Transversality and Lefschetz numbers for foliation maps

Let $F$ be a smooth foliation on a closed Riemannian manifold $M$, and let $Λ$ be a transverse invariant measure of $F$. Suppose that $Λ$ is absolutely continuous with respect to the Lebesgue measure on smooth transversals. Then a topological definition of the $Λ$-Lefschetz number of any leaf preserving diffeomorphism $(M,F)\to(M,F)$ is given. For this purpose, standard results about smooth approximation and transversality are extended to the case of foliation maps. It is asked whether this topological $Λ$-Lefschetz number is equal to the analytic $Λ$-Lefschetz number defined by Heitsch and La…

3维流形中横截叶状结构的交：Hausdorff叶空间蕴含叶状拟测地线

Intersection of transverse foliations in 3-manifolds: Hausdorff leafspace implies leafwise quasi-geodesic

本文研究闭3维流形中两横截二维叶状结构相交所产生的一维叶状结构，证明其叶状拟测地性质等价于万有覆盖中诱导叶状结构具有 Hausdorff 叶空间。

Basic automorphism group of complete Cartan foliations covered by fibration

We prove a theorem that gives a sufficient condition for the full basic automorphism group of a complete Cartan foliation to admit a unique (finite-dimensional) Lie group structure in the category of Cartan foliations. Emphasize that the transverse Cartan geometry may not be effective. Some estimates of the dimension of this group depending on the transverse geometry are found. Further, we investigate Cartan foliations covered by fibrations. When the global holonomy group of that foliation is discrete, we obtain the explicit new formula for determining its basic automorphism Lie group. Example…

Transverse Kähler structures on central foliations of complex manifolds

For a compact complex manifold, we introduce holomorphic foliations associated with certain abelian subgroups of the automorphism group. Such foliations are generalizations of holomorphic principal torus bundles. If there exists a transverse Kähler structure on such a foliation, then we obtain a nice differential graded algebra which is quasi-isomorphic to the de Rham complex and a nice differential bigraded algebra which is quasi-isomorphic to the Dolbeault complex like the formality of compact Kähler manifolds. Moreover, under certain additional condition, we can develop Morgan's theory of m…

Left orderability, foliations, and transverse $(π_1,\mathbb{R})$ structures for $3$-manifolds with sphere boundary

Let $M$ be a closed orientable irreducible $3$-manifold such that $π_1(M)$ is left orderable. (a) Let $M_0 = M - Int(B^{3})$, where $B^{3}$ is a compact $3$-ball in $M$. We have a process to produce a co-orientable Reebless foliation $\mathcal{F}$ in $M_0$ such that: (1) $\mathcal{F}$ has a transverse $(π_1(M),\mathbb{R})$ structure, (2) there exists a simple closed curve in $M$ that is co-orientably transverse to $\mathcal{F}$ and intersects every leaf of $\mathcal{F}$. More specifically, given a pair $(<,Γ)$ composed of a left-invariant order "$<$" of $π_1(M)$ and a fundamental domain $Γ$ of…

D-模与可解李叶状结构

D-modules and Solvable Lie Foliations

本文从D-模理论出发研究紧致流形上的可解李叶状结构，通过计算三类叶状结构的D-不规则度，证明和乐群非多循环性等价于导出环高阶上同调非消失，建立几何与D-模不变量的新联系。

横截于叶状结构的Reeb流

Reeb flows transverse to foliations

本文证明了闭有向3-流形上的共向$C^2$叶状结构可被扰动为具有横截Reeb流的切触结构当且仅当其不支持不变横截测度，并利用逐叶布朗运动给出了Eliashberg-Thurston定理的新视角。

强制闭轨道的叶状结构

Foliations forcing closed orbits

本文刻画了可环3维流形上C²有限深度叶状结构中强制所有横截流存在闭轨道的性质，利用周期端映射构造闭轨道，还通过插入动力插件构造出无闭轨道的C¹流。

$C^{1,0}$ Foliation Theory

Transverse one dimensional foliations play an important role in the study of codimension one foliations. In \cite{KR2}, the authors introduced the notion of flow box decomposition of a 3-manifold $M$. This is a decomposition of $M$ that reflects both the structure of a given codimension one foliation and that of a given transverse flow. In this paper, flow box decompositions are used to extend some classical foliation results to foliations that are not $C^2$. Enhancements of well-known results of Calegari on smoothing leaves, Dippolito on Denjoy blowup of leaves, and Tischler on approximations…

Seifert $3$-流形上横截叶层的调和测度与刚性

Harmonic measures and rigidity for transverse foliations on Seifert $3$-manifolds

本文研究Seifert 3-流形上横截叶层的调和测度，证明Gauss-Bonnet公式及最大Euler数调和测度的刚性，并推导出叶层的半共轭刚性定理。

深度一叶状结构、伪Anosov流和万有圆

Depth-one foliations, pseudo-Anosov flows and universal circles

本文证明在闭无环面三维流形中，拉紧深度一叶状结构相关的万有圆同构于无完美拟合伪Anosov流的理想边界，并推论该横截伪Anosov流在轨道等价下至多唯一。

Codimension one minimal foliations whose leaves have fundamental groups with the same polynomial growth

Let $\mathcal{F}$ be a transversely orientable codimension one minimal foliation without vanishing cycles of a manifold $M$. We show that if the fundamental group of each leaf of $\mathcal{F}$ has polynomial growth of degree $k$ for some non-negative integer $k$, then the foliation $\mathcal{F}$ is without holonomy.

Promoting Essential Laminations

We show that every co--orientable taut foliation F of an orientable, atoroidal 3-manifold admits a transverse essential lamination. If this transverse lamination is a foliation G, the pair F,G are the unstable and stable foliation respectively of an Anosov flow. Otherwise, F admits a pair of transverse very full genuine laminations. In the second case, M satisfies the weak geometrization conjecture - either its fundamental group contains Z+Z or it is word-hyperbolic. Moreover, if M is atoroidal, the mapping class group of M is finite, and any automorphism homotopic to the identity is isotopic …

Variation of the Liouville measure of a hyperbolic surface

For a compact riemannian manifold of negative curvature, the geodesic foliation of its unit tangent bundle is independent of the negatively curved metric, up to Holder bicontinuous homeomorphism. However, the riemannian metric defines a natural transverse measure to this foliation, the Liouville transverse measure, which does depend on the metric. For a surface S, we show that the map which to a hyperbolic metric on S associates its Liouville transverse measure is differentiable, in an appropriate sense. Its tangent map is valued in the space of transverse Holder distributions for the geodesic…

三维部分双曲微分同胚、遍历性与横截叶状结构

Partially hyperbolic diffeormorphisms, ergodicity, and transverse foliations in dimension 3

本文完成Pujals计划，基于Anosov流给出三维流形中（链）传递部分双曲微分同胚的完整拓扑分类，解答Hertz-Hertz-Ures的遍历性猜想，其定理B将在后续文章中推广以移除链传递假设。