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The current paper is devoted to the asymptotic behavior of a class of stochastic PDE. More precisely, with the help of the theory of integrated semigroups and a crucial estimate of the random Stieltjes convolution, we study the existence and smoothness of center-unstable invariant manifolds and center-stable foliations for a class of stochastic PDE with non-dense domain through the Lyapunov-Perron method. Finally, we give two examples, a stochastic age-structured model and a stochastic parabolic equation, to illustrate our results. [ABSTRACT FROM AUTHOR]

This is a survey article, with essentially complete proofs, of a series of recent results concerning the geometry of the characteristic foliation on smooth divisors in compact hyperkähler manifolds, starting with work by Hwang–Viehweg, but also covering articles by Amerik–Campana and Abugaliev. The restriction of the holomorphic symplectic form on a hyperkähler manifold X$X$ to a smooth hypersurface D⊂X$D\subset X$ leads to a regular foliation F⊂TD${\mathcal {F}}\subset {\mathcal {T}}_D$ of rank 1, the characteristic foliation. The picture is complete in dimension 4 and shows that the behaviour of the leaves of F${\mathcal {F}}$ on D$D$ is determined by the Beauville–Bogomolov square q(D)$q(D)$ of D$D$. In higher dimensions, some of the results depend on the abundance conjecture for D$D$. [ABSTRACT FROM AUTHOR]

We study \mathbb {R}^k \times \mathbb {Z}^\ell actions on arbitrary compact manifolds with a projectively dense set of Anosov elements and 1-dimensional coarse Lyapunov foliations. Such actions are called totally Cartan actions. We completely classify such actions as built from low-dimensional Anosov flows and diffeomorphisms and affine actions, verifying the Katok-Spatzier conjecture for this class. This is achieved by introducing a new tool, the action of a dynamically defined topological group describing paths in coarse Lyapunov foliations, and understanding its generators and relations. We obtain applications to the Zimmer program. [ABSTRACT FROM AUTHOR]

We show that any noncompact oriented surface is homeomorphic to the leaf of a minimal foliation of a closed 3-manifold. These foliations are (or are covered by) suspensions of continuous minimal actions of surface groups on the circle. Moreover, the above result is also true for any prescription of a countable family of topologies of noncompact surfaces: they can coexist in the same minimal foliation. All the given examples are hyperbolic foliations, meaning that they admit a leafwise Riemannian metric of constant negative curvature. Many oriented Seifert manifolds with a fibered incompressible torus and whose associated orbifold is hyperbolic admit minimal foliations as above. The given examples are not transversely C²-smoothable. [ABSTRACT FROM AUTHOR]

We study the existence of 3-2-1 foliations adapted to Reeb flows on the tight 3-sphere. These foliations admit precisely three binding orbits whose Conley-Zehnder indices are 3, 2, and 1, respectively. All regular leaves are disks and annuli asymptotic to the binding orbits. Our main results provide sufficient conditions for the existence of 3-2-1 foliations with prescribed binding orbits. We also exhibit a concrete Hamiltonian on \mathbb {R}^4 admitting 3-2-1 foliations when restricted to suitable energy levels. [ABSTRACT FROM AUTHOR]

In this paper, we show the existence of uniform rational lc polytopes for foliations with functional boundaries in dimension ⩽3$\leqslant 3$. As an application, we prove the global ACC for foliated threefolds with arbitrary DCC coefficients. We also provide applications on the accumulation points of lc thresholds of foliations in dimension ⩽3$\leqslant 3$. [ABSTRACT FROM AUTHOR]

For a foliation ℱ of a smooth complex manifold, we introduce the category of V -structures associated to a vertex operator algebra V and the category of its modules. The main result consists of the construction of V -structures and canonicity proof of on ℱ. [ABSTRACT FROM AUTHOR]

In this paper, we study slant Riemannian submersions from conformal Sasakian manifolds onto Riemannian manifolds. We investigate the geometry of foliations associated with the submersion and give sufficient conditions for the submersion to be harmonic. [ABSTRACT FROM AUTHOR]

This paper is a part of a series devoted to the Euclidean-hyperboloidal foliation method introduced by the authors for investigating the global existence problem associated with nonlinear systems of coupled wave-Klein–Gordon equations with small data. This method was developed especially for investigating the initial value problem for the Einstein-massive field system in wave gauge. Here, we study the (fourth-order) field equations of f(R) modified gravity and investigate the global dynamical behavior of the gravitational field in the near-Minkowski regime. We establish the existence of a globally hyperbolic Cauchy development approaching Minkowski spacetime (in spacelike, null, and timelike directions), when the initial data set is sufficiently close to an asymptotically Euclidean and spacelike hypersurface in Minkowski spacetime. We cast the (fourth-order) f(R)-field equations in the form of a second-order wave-Klein–Gordon system, which has an analogous structure to the Einstein-massive field system but, in addition, involves a (possibly small) effective mass parameter. We establish the nonlinear stability of the Minkowski spacetime in the context of f(R) gravity, when the integrand f(R) in the action functional can be taken to be arbitrarily close to the integrand R of the standard Hilbert–Einstein action. [ABSTRACT FROM AUTHOR]

Let F be a smooth Riemann surface foliation on M \ E , where M is a complex manifold and the singular set E ⊂ M is an analytic set of codimension at least two. Fix a hermitian metric on M and assume that all leaves of F are hyperbolic. Verjovsky's modulus of uniformization η is a positive real function defined on M \ E defined in terms of the family of holomorphic maps from the unit disc D into the leaves of F and is a measure of the largest possible derivative in the class of such maps. Various conditions are known that guarantee the continuity of η on M \ E . The main question that is addressed here is its continuity at points of E. To do this, we adapt Whitney's C 4 -tangent cone construction for analytic sets to the setting of foliations and use it to define the tangent cone of F at points of E. This leads to the definition of a foliation that is of transversal type at points of E. It is shown that the map η associated to such foliations is continuous at E provided that it is continuous on M \ E and F is of transversal type. We also present observations on the locus of discontinuity of η . Finally, for a domain U ⊂ M , we consider F U , the restriction of F to U and the corresponding positive function η U . Using the transversality hypothesis leads to strengthened versions of the results of Lins Neto–Martins on the variation U ↦ η U . [ABSTRACT FROM AUTHOR]

For any closed orientable 3‐manifold, there is a volume function defined on the space of all Seifert representations of the fundamental group. The maximum absolute value of this function agrees with the Seifert volume of the manifold due to Brooks and Goldman. For any Seifert representation of a graph manifold, the authors establish an effective formula for computing its volume, and obtain restrictions to the representation as analogous to the Milnor–Wood inequality (about transversely projective foliations on Seifert fiber spaces). It is shown that the Seifert volume of any graph manifold is a rational multiple of π2$\pi ^2$. Among all finite covers of a given nongeometric graph manifold, the supremum ratio of the Seifert volume over the covering degree can be a positive number, and can be infinite. Examples of both possibilities are discovered, and confirmed with the explicit values determined for the finite ones. [ABSTRACT FROM AUTHOR]

Let |$Y$| be a smooth hypersurface in a projective irreducible holomorphic symplectic manifold X of dimension 2n. The characteristic foliation |$F$| is the kernel of the symplectic form restricted to Y. In this article, we prove that a generic leaf of the characteristic foliation is dense in Y if Y has positive Beauville–Bogomolov–Fujiki square. [ABSTRACT FROM AUTHOR]

We consider a class of endomorphisms that contains a set of piecewise partially hyperbolic dynamics semi-conjugated to non-uniformly expanding maps. Our goal is to study a class of endomorphisms that preserve a foliation that is almost everywhere uniformly contracted, with possible discontinuity sets parallel to the contracting direction. We apply the spectral gap property and the $ \zeta $-Hölder regularity of the disintegration of its equilibrium states to prove a quantitative statistical stability statement. More precisely, under deterministic perturbations of the system of size $ \delta $, we show that the $ F $-invariant measure varies continuously with respect to a suitable anisotropic norm. Furthermore, we establish that certain interesting classes of perturbations exhibit a modulus of continuity estimated by $ D_2\delta^\zeta \log \delta $, where $ D_2 $ is a constant. [ABSTRACT FROM AUTHOR]

We study foliations by chord–arc Jordan curves of the twice punctured Riemann sphere C∖{0}$\mathbb {C} \setminus \lbrace 0\rbrace$ using the Loewner–Kufarev equation. We associate to such a foliation a function on the plane that describes the "local winding" along each leaf. Our main theorem is that this function has finite Dirichlet energy if and only if the Loewner driving measure ρ$\rho$ has finite Loewner–Kufarev energy, defined by S(ρ)=12∫∫S1×Rνt′(θ)2dθdt$$\begin{equation*} \hspace*{58pt}S(\rho) = \frac{1}{2}\iint\nolimits _{S^1 \times \mathbb {R}} \nu _t^{\prime }(\theta)^2 \, d \theta d t \end{equation*}$$whenever ρ$\rho$ is of the form νt(θ)2dθdt$\nu _t(\theta)^2 d \theta d t$, and set to ∞$\infty$ otherwise. Moreover, if either of these two energies is finite, they are equal up to a constant factor, and in this case, the foliation leaves are Weil–Petersson quasicircles. This duality between energies has several consequences. The first is that the Loewner–Kufarev energy is reversible, that is, invariant under inversion and time reversal of the foliation. Furthermore, the Loewner energy of a Jordan curve can be expressed using the minimal Loewner–Kufarev energy of those measures that generate the curve as a leaf. This provides a new and quantitative characterization of Weil–Petersson quasicircles. Finally, we consider conformal distortion of the foliation and show that the Loewner–Kufarev energy satisfies an exact transformation law involving the Schwarzian derivative. The proof of our main theorem uses an isometry between the Dirichlet energy space on the unit disc and L2(2ρ)$L^2(2\rho)$ that we construct using Hadamard's variational formula expressed by means of the Loewner–Kufarev equation. Our results are related to κ$\kappa$‐parameter duality and large deviations of Schramm–Loewner evolutions coupled with Gaussian random fields. [ABSTRACT FROM AUTHOR]

In this work we derive explicit entropy bounds for two classes of closed self-shrinkers: the class of embedded closed self-shrinkers recently constructed in Riedler (in J Geom Anal 33(6):Paper No. 172, 2023) using isoparametric foliations of spheres, and the class of compact non-spherical immersed rotationally symmetric self-shrinkers. These bounds generalize the entropy bounds found in Ma, Muhammad, Møller (in J Reine Angew Math 793:239—259, 2022) on the space of complete embedded rotationally symmetric self-shrinkers. [ABSTRACT FROM AUTHOR]

The use of ideal memristors in a continuous-time (CT) nonlinear circuit is known to greatly enrich the dynamic behavior with respect to the memristorless counterpart, which is a crucial property for applications in future analog electronic circuits. This can be explained via the flux–charge analysis method (FCAM), according to which CT circuits with ideal memristors have for structural reasons first integrals (or invariants of motion, or conserved quantities) and their state space can be foliated in infinitely many invariant manifolds where they can display different dynamics. The paper introduces a new discretization scheme for the memristor which, differently from those adopted in the literature, guarantees that the first integrals of the CT memristor circuits are preserved exactly in the discretization, and that this is true for any step size. This new scheme makes it possible to extend FCAM to discrete-time (DT) memristor circuits and rigorously show the existence of invariant manifolds and infinitely many coexisting attractors (extreme multistability). Moreover, the paper addresses standard bifurcations varying the discretization step size and also bifurcations without parameters, i.e. bifurcations due to varying the initial conditions for fixed step size and circuit parameters. The method is illustrated by analyzing the dynamics and flip bifurcations with and without parameters in a DT memristor–capacitor circuit and the Poincaré–Andronov–Hopf bifurcation in a DT Murali–Lakshmanan–Chua circuit with a memristor. Simulations are also provided to illustrate bifurcations in a higher-order DT memristor Chua's circuit. The results in the paper show that DT memristor circuits obtained with the proposed discretization scheme are able to display even richer dynamics and bifurcations than their CT counterparts, due to the coexistence of infinitely many attractors and the possibility to use the discretization step as a parameter without destroying the foliation in invariant manifolds. [ABSTRACT FROM AUTHOR]

Let F be a transversally oriented codimension-one foliation of class Cr, r ≥ 0, on a closed manifold M. A leaf class of a leaf F is the union of all leaves having the same closure as F. Let X be the leaf classes space and X0 be the union of all open subsets of X homeomorphic to R or S1. In [3, Theorem 3.15] it is shown that if a codimension one foliation has a finite height, then the singular part of the space of leaf classes is homeomorphic to the prime spectrum (or simply the spectrum) of unitary commutative ring. In this paper we prove that the singular part of the space of leaf classes is homeomorphic to the spectrum of unitary commutative ring if and only if every family of totaly ordered leaves is bounded below. [ABSTRACT FROM AUTHOR]

In this paper, we establish a number of results about the topology of the leaves of a closed singular Riemannian foliation .M;F /. If M is simply connected, we prove that the leaves are finitely covered by nilpotent spaces, and characterize the fundamental group of the generic leaves. If M has virtually nilpotent fundamental group, we prove that the leaves have virtually nilpotent fundamental group as well. [ABSTRACT FROM AUTHOR]

An embedded curve in a symplectic surface Σ⊂X$\Sigma \subset X$ defines a smooth deformation space B$\mathcal {B}$ of nearby embedded curves. A key idea of Kontsevich and Soibelman is to equip the symplectic surface X$X$ with a foliation in order to study the deformation space B$\mathcal {B}$. The foliation, together with a vector space VΣ$V_\Sigma$ of meromorphic differentials on Σ$\Sigma$, endows an embedded curve Σ$\Sigma$ with the structure of the initial data of topological recursion, which defines a collection of symmetric tensors on VΣ$V_\Sigma$. Kontsevich and Soibelman define an Airy structure on VΣ$V_\Sigma$ to be a formal quadratic Lagrangian L⊂T∗(VΣ∗)$\mathcal {L}\subset T^*(V_\Sigma ^*)$ which leads to an alternative construction of the tensors of topological recursion. In this paper, we produce a formal series θ$\theta$ on B$\mathcal {B}$ which takes it values in L$\mathcal {L}$, and use this to produce the Donagi–Markman cubic from a natural cubic tensor on VΣ$V_\Sigma$, giving a generalisation of a result of Baraglia and Huang. [ABSTRACT FROM AUTHOR]

We investigate the geometry of universal embedding spaces for compact almost-complex manifolds of a given dimension, and related constructions that allow for an extrinsic study of the integrability of almost-complex structures. These embedding spaces were introduced by J-P. Demailly and H. Gaussier, and are complex algebraic analogues of twistor spaces. Their goal was to study a conjecture made by F. Bogomolov asserting the "transverse embeddability" of arbitrary compact complex manifolds into foliated algebraic varieties. In this work, we introduce a more general category of universal embedding spaces, and elucidate the geometric structure of related bundles, such as the integrability locus characterizing integrable almost-complex structures. Our approach could potentially lead to finding new obstructions to the existence of a complex structure, which may be useful for tackling Yau's Challenge. [ABSTRACT FROM AUTHOR]

The PSL (4 , R) Hitchin component of a closed surface group π 1 (S) consists of holonomies of properly convex foliated projective structures on the unit tangent bundle of S. We prove that the leaves of the codimension-1 foliation of any such projective structure are all projectively equivalent if and only if its holonomy is Fuchsian. This implies constraints on the symmetries and shapes of these leaves. We also give an application to the topology of the non- T 0 space C (R P n) of projective classes of properly convex domains in R P n. Namely, Benzécri asked in 1960 if every closed subset of C (R P n) that contains no proper nonempty closed subset is a point. Our results imply a negative resolution for n ≥ 2. [ABSTRACT FROM AUTHOR]

We study the problem of classifying local projective structures in dimension two having non trivial Lie symmetries. In particular we obtain a classification of foliated projective structures having positive dimensional Lie algebra of projective vector fields. [ABSTRACT FROM AUTHOR]

Given a function f on a smooth Riemannian manifold without boundary, we prove that if p ∈ M is a non-degenerate critical point of f , then a neighborhood of p contains a foliation by spheres with mean curvature proportional to f. This foliation is essentially unique. The nondegeneracy assumption can be substantially relaxed, at the expense of losing the property that the family of spheres with prescribed mean curvature defines a foliation. [ABSTRACT FROM AUTHOR]

In this paper, we discuss how to travel along horizontal broken geodesics of a homogeneous Finsler submersion, i.e., we study, what in Riemannian geometry was called by Wilking, the dual leaves. More precisely, we investigate the attainable sets A q (C) of the set of analytic vector fields C determined by the family of horizontal unit geodesic vector fields C to the fibers F = { ρ − 1 (c) } of a homogeneous analytic Finsler submersion ρ : M → B. Since reverse of geodesics don't need to be geodesics in Finsler geometry, one can have examples on non compact Finsler manifolds M where the attainable sets (the dual leaves) are no longer orbits or even submanifolds. Nevertheless we prove that, when M is compact and the orbits of C are embedded, then the attainable sets coincide with the orbits. Furthermore, if the flag curvature is positive then M coincides with the attainable set of each point. In other words, fixed two points of M , one can travel from one point to the other along horizontal broken geodesics. In addition, we show that each orbit O (q) of C associated to a singular Finsler foliation coincides with M , when the flag curvature is positive, i.e., we prove Wilking's result in Finsler context. In particular we review Wilking's transversal Jacobi fields in Finsler case. [ABSTRACT FROM AUTHOR]

The idea of Lichnerowicz or Morse-Novikov cohomology groups of a manifold has been utilized by many researchers to study important properties and invariants of a manifold. Morse-Novikov cohomology is defined using the differential d ω = d + ω ∧ , where ω is a closed 1-form. We study Morse-Novikov cohomology relative to a foliation on a manifold and its homotopy invariance and then extend it to more general type of forms on a Riemannian foliation. We study the Laplacian and Hodge decompositions for the corresponding differential operators on reduced leafwise Morse-Novikov complexes. In the case of Riemannian foliations, we prove that the reduced leafwise Morse-Novikov cohomology groups satisfy the Hodge theorem and Poincaré duality. The resulting isomorphisms yield a Hodge diamond structure for leafwise Morse-Novikov cohomology. [ABSTRACT FROM AUTHOR]

A stratified space is a kind of topological space together with a partition into smooth manifolds. These kinds of spaces naturally arise in the study of singular algebraic varieties, symplectic reduction, and differentiable stacks. In this paper, we introduce a particular class of stratified spaces called stratified vector bundles, and provide an alternate characterization in terms of monoid actions. We will then provide large families of examples coming from the theory of Whitney stratified spaces, singular foliation theory, and equivariant vector bundle theory. Finally, we extend functorial properties of smooth vector bundles to the stratified case. [ABSTRACT FROM AUTHOR]

We study the twisted Bott-Chern and Aeppli cohomologies on transversely Hermitian foliations. And we study the Frölicher type inequality of these twisted cohomologies. As an application, we prove the Hodge isomorphism on transverse Kähler foliations. [ABSTRACT FROM AUTHOR]

Originally, the noncommutative residue was studied in the 80's by Wodzicki in his thesis [33] and Guillemin [19]. In this article we give a definition of the Wodzicki residue, using the language of r -fibered distributions from [24] , [30] , in the context of filtered manifolds. We show that this groupoidal residue behaves like a trace on the algebra of pseudodifferential operators on filtered manifolds and coincides with the usual residue Wodzicki in the case where the manifold is trivially filtered. Moreover, in the context of Heisenberg calculus, we show that the groupoidal residue coincides with Ponge's definition [25] for contact and codimension 1 foliation Heisenberg manifolds. [ABSTRACT FROM AUTHOR]

We prove that any holomorphic vector bundle admitting a holomorphic connection, over a compact Kähler Calabi-Yau manifold, also admits a flat holomorphic connection. This addresses a particular case of a question asked by Atiyah and generalizes a result previously obtained in [6] for simply connected compact Kähler Calabi-Yau manifolds. We give some applications of it in the framework of Cartan geometries and foliated Cartan geometries on Kähler Calabi-Yau manifolds. [ABSTRACT FROM AUTHOR]