Nataliya Goncharuk, Yury Kudryashov
In this article, we prove two results. First, we construct a dense subset in the space of polynomial foliations of degree $n$ such that each foliation from this subset has a leaf with at least $\frac{(n+1)(n+2)}2-4$ handles. Next, we prove that for a generic foliation invariant under the map $(x, y)\mapsto (x, -y)$ all leaves have infinitely many handles.
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We study conformal structure and topology of leaves of singular foliations by Riemann surfaces.
We study topology of leaves of 1-dimensional singular holomorphic foliations of Stein manifolds. We prove that for a generic foliation all leaves, except for at most countably many, are contractible, the rest are topological cylinders. We show that a generic foliation is complex Kupka-Smale.
We describe the topological types of leaves of generic logarithmic foliations on the complex projective plane. We prove that all leaves, except for a finite many are biholomorphic to $\mathbb{C}$ or homeomorphic to the surface known as Loch Ness monster.
In 2001 E. Ghys, X. Gomez-Mont and J. Saludes defined the Fatou and Julia components of transversely holomorphic foliations on compact manifolds. It is a partition of the manifold in two saturated sets: the Fatou set which represents the non-chaotic part of the foliation and its complementary, the Julia set. Using the Brownian motion transverse to the foliation, it is proved in this paper that, if the foliation is taut and if F is a wandering component of the Fatou set, then almost every point of the topological boundary of F (almost for any harmonic measure on the boundary) is a limit point o…
We discuss various problems regarding the structure of the foliation of some foliated submanifolds S of C^n, in particular Levi flat ones. As a general scheme, we suppose that S is bounded along a coordinate (or a subset of coordinates), and prove that the leaves of its foliation are complex planes.
Let X be a compact complex surface with a real foliation. If all leaves are compact complex curves, the foliation must be holomorphic.
We study those real $\mathcal{C}^\infty$ foliations in complex surfaces whose leaves are holomorphic curves. The main motivation is to try and understand these foliations in neighborhoods of curves: can we expect the space of foliations in a fixed neighborhood to be infinite-dimensional, or are there some contexts under which every such foliation is holomorphic? We give some restrictions and study in more details the geometry of foliations whose leaves belong to a holomorphic family of holomorphic curves. In particular, we classify all real-analytic foliations on neighborhoods of curves which …
Given a smooth foliation by complex curves (locally around a point $x\in\mathbb{C}^2\setminus\{0\}$) which is "compatible" with the foliation by spheres centered at the origin, we construct a smooth real-valued function $g$ in a neighborhood of said point, which is positive, homogeneous and constant along the leaves. A corollary we obtain from this is relevant to the problem of "bumping out" certain pseudoconvex domains in $\mathbb{C}^3$.
We study the homotopy groups of generic leaves of logarithmic foliations on complex projective manifolds. We exhibit a relation between the homotopy groups of a generic leaf and of the complement of the polar divisor of the logarithmic foliation.
We consider closed positive currents invariant by a singular holomorphic foliation on an algebraic surface. We show that under some conditions the foliation must leave invariant an algebraic curve.
We consider foliations of compact complex manifolds by analytic curves. We suppose that the line bundle tangent to the foliation is negative. We show that in a generic case there exists a finitely smooth homeomophism, holomorphic on the fibers and mapping fiberwise the manifold of universal coverings over the leaves passing through some transversal $B$ onto some domain in $B\times\mathbb{C}$ with continuous boundary. depending on the leaves. The problem can be reduced to a study of the Beltrami equation with parameters on the unit disk in the case, when derivatives of the corresponding coeffic…
Stability of closed leaves holomorphic foliations via torsion behavior of groups of germs
本文通过引入胚群扭转轨迹，证明了闭叶全纯叶状结构中存在高维紧致叶不变解析集，且叶体积一致有界，并给出射影情形的全局化。
In this work we consider holomorphic foliations of degree two on the projective plane $\mathbb{P}^2$ having an invariant line. In a suitable choice of affine coordinates these foliations are induced by a quadratic vector field over the affine part in such a way that the invariant line corresponds to the line at infinity. We say that two such foliations are topologically equivalent provided there exists a homeomorphism of $\mathbb{P}^2$ which brings the leaves of one foliation onto the leaves of the other and preserves orientation both on the ambient space and on the leaves. The main result of …
In this paper, we study a notion of hyperbolicity for hyperbolicity foliations with 1-dimensional parabolic leaves, namely the non-existence of holomorphic cylinders along the foliation - holomorphic maps from $\D^{n-1} \times \C$ to the manifold sending each $\{*\} \times \C$ to a leaf. We construct a tensor, that is a leaf-wise holomorphic section of a bundle above the foliated manifold, which vanishes on an open saturated set if and only if there exists such a cylinder through each point of this open set. Thanks to this study, we prove that open sets saturated by compact leaves are union of…
A singular real analytic foliation $\mathcal{F}$ of real codimension one on an $n$-dimensional complex manifold $M$ is Levi-flat if each of its leaves is foliated by immersed complex manifolds of dimension $n-1$. These complex manifolds are leaves of a singular real analytic foliation $\mathcal{L}$ which is tangent to $\mathcal{F}$. In this article, we classify germs of Levi-flat foliations at $(\mathbb{C}^{n},0)$ under the hypothesis that $\mathcal{L}$ is a germ holomorphic foliation. Essentially, we prove that there are two possibilities for $\mathcal{L}$, from which the classification of $\…
Holomorphic Linear $\C^k$-Actions, Trace Foliations, and Higher-Rank Poincaré Dynamics
本文研究高阶框架下对角全纯$\boldsymbol{\rm C}^k$-作用生成的轨道分解，通过坐标分层分析叶的结构，揭示其微分同胚类型的阈值现象，并证明奇异叶状结构具有典范局部横向全纯结构。
We study non-compact surfaces obtained by gluing strips $\mathbb{R}\times(-1,1)$ with at most countably many boundary intervals along some these intervals. Every such strip possesses a foliation by parallel lines, which gives a foliation on the resulting surface. It is proved that the identity path component of the group of homeomorphisms of that foliation is contractible.
Let $X$ be a connected non-compact $2$-dimensional manifold possibly with boundary and $Δ$ be a foliation on $X$ such that each leaf $ω\inΔ$ is homeomorphic to $\mathbb{R}$ and has a trivially foliated neighborhood. Such foliations on the plane were studied by W. Kaplan who also gave their topological classification. He proved that the plane splits into a family of open strips foliated by parallel lines and glued along some boundary intervals. However W. Kaplan's construction depends on a choice of those intervals, and a foliation is described in a non-unique way. We propose a canonical cuttin…
We investigate the accumulation to singular points of leaves of codimension one foliations whose normal bundle is ample, with emphasis on the nonexistence of Levi-flat hypersurfaces.
A semiholomorphic foliations of type (n, d) is a differentiable real manifold X of dimension 2n + d, foliated by complex leaves of complex dimension n. In the present work, we introduce an appropriate notion of pseudoconvexity (and consequently, q-completeness) for such spaces, given by the interplay of the usual pseudoconvexity, along the leaves, and the positivity of the transversal bundle. For 1-complete real analytic semiholomorphic foliations, we obtain a vanishing theorem for the CR cohomology, which we use to show an extension result for CR functions on Levi flat hypersurfaces and an em…
Wild holomorphic foliations of the ball
该研究证明开单位球存在由闭复超曲面构成的非奇异全纯叶状结构，其完全叶与不完全叶的并集均稠密，首次给出此类含完全叶为不完全叶极限的全纯叶状结构示例，还得到任意纯余维子流形叶状结构的类似结果。
We prove that a transversely holomorphic foliation which is transverse to the fibers of a fibration, is a Seifert fibration if the set of compact leaves is not of zero measure. Similarly, we prove that a finitely generated subgroup of holomorphic diffeomorphisms of a connected complex manifold, is finite provided that the set of periodic orbits is not of zero measure.
Errett Bishop theorems on Complex Analytic Sets: Chow's Theorem Revisited and Foliations with all leaves Compact on Kähler Manifolds
本文利用 Bishop 关于解析簇体积与极限的结果统一重证 Chow、Remmert–Stein、Montel 及 Edwards–Millet–Sullivan 等复分析与叶状结构定理。
Let $Z$ be a non-compact two-dimensional manifold obtained from a family of open strips $\mathbb{R}\times(0,1)$ with boundary intervals by gluing those strips along their boundary intervals. Every such strip has a foliation into parallel lines $\mathbb{R}\times t$, $t\in(0,1)$, and boundary intervals, whence we get a foliation $Δ$ on all of $Z$. Many types of foliations on surfaces with leaves homeomorphic to the real line have such "striped" structure. That fact was discovered by W. Kaplan (1940-41) for foliations on the plane $\mathbb{R}^2$ by level-set of pseudo-harmonic functions $\mathbb{…
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