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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]

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]

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 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]

Suppose that C0n⊂Rn+1$\mathbf {C}_0^n \subset \mathbb {R}^{n+1}$ is a smooth strictly minimizing and strictly stable minimal hypercone (such as the Simons cone), l≥0$l \ge 0$, and M$M$ a complete embedded minimal hypersurface of Rn+1+l$\mathbb {R}^{n+1+l}$ lying to one side of C=C0×Rl$\mathbf {C}= \mathbf {C}_0 \times \mathbb {R}^l$. If the density at infinity of M$M$ is less than twice the density of C$\mathbf {C}$, then we show that M=H(λ)×Rl$M = H(\lambda) \times \mathbb {R}^l$, where {H(λ)}λ$\lbrace H(\lambda)\rbrace _\lambda$ is the Hardt–Simon foliation of C0$\mathbf {C}_0$. This extends a result of L. Simon, where an additional smallness assumption is required for the normal vector of M$M$. [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 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]

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]

We prove that the set of leaves of a singular foliation with the Nishimori relation is partially ordered if and only if all leaves are proper. [ABSTRACT FROM AUTHOR]

Bill Thurston proved that taut foliations of hyperbolic 3‐manifolds have Euler classes of norm at most one, and conjectured that any integral second cohomology class of norm equal to one is realized as the Euler class of some taut foliation. Recent work of the second author, joint with David Gabai, has produced counterexamples to this conjecture. Since tight contact structures exist whenever taut foliations do and their Euler classes also have norm at most one, it is natural to ask whether the Euler class one conjecture might still be true for tight contact structures. In this paper, we show that the previously constructed counterexamples for Euler classes of taut foliations in Mehdi Yazdi [Acta Math. 225 (2020) no. 2, 313–368] are in fact realized as Euler classes of tight contact structures. This provides some evidence for the Euler class one conjecture for tight contact structures. [ABSTRACT FROM AUTHOR]

In this article we study Stäckel representations of stationary KdV systems. Using Lax formalism we prove that these systems have two different representations as separable Stäckel systems of Benenti type, related with different foliations of the stationary manifold. We do it by constructing an explicit transformation between the jet coordinates of stationary KdV systems and separation variables of the corresponding Benenti systems for arbitrary number of degrees of freedom. Moreover, on the stationary manifold, we present the explicit form of Miura map between both representations of stationary KdV systems, which also yields their bi-Hamiltonian formulation. [ABSTRACT FROM AUTHOR]

Here we prove that the dynamics on any two center-manifolds of a fixed point of any C k , 1 dynamical system of finite dimension with k ≥ 1 are C k -conjugate to each other. For pedagogical purpose, we also extend Perron's method for differential equations to diffeomorphisms to construct the theory of invariant manifolds and invariant foliations at fixed points of dynamical systems of finite dimensions. [ABSTRACT FROM AUTHOR]

In this paper, we prove the rational coefficient case of the global ACC for foliated threefolds. Specifically, we consider any lc foliated log Calabi-Yau triple (X,\mathcal {F},B) of dimension 3 whose coefficients belong to a set \Gamma of rational numbers satisfying the descending chain condition, and prove that the coefficients of B belong to a finite set depending only on \Gamma. To prove our main result, we introduce the concept of generalized foliated quadruples, which is a mixture of foliated triples and Birkar-Zhang's generalized pairs. With this concept, we establish a canonical bundle formula for foliations in any dimension. As for applications, we extend Shokurov's global index conjecture in the classical MMP to foliated triples and prove this conjecture for threefolds with nonzero boundaries and for surfaces. Additionally, we introduce the theory of rational polytopes for functional divisors on foliations and prove some miscellaneous results. [ABSTRACT FROM AUTHOR]

Lins Neto [Ann. Sci. École Norm. Sup. (4) 35 (2002), pp. 231–266] constructed families of foliations which are counterexamples to Poincaré's Problem and Painlevé's Problem. We will determine the minimal models of these families of foliations, calculate their Chern numbers, Kodaira dimension, and numerical Kodaira dimension. We prove that the slopes of Lins Neto's foliations are at least 6, and their limits are bigger than 7. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a certain topology on the group DiffF (M) of all Cr-diffeomorphisms of the foliated manifold (M; F), where r ≥ 0. This topology depends on the foliation and is called the F-compact-open topology. It coincides with the compact-open topology when F is an n-dimensional foliation. If the codimension of the foliation is n, then the convergence in this topology coincides with the pointwise convergence, where n = dim M. We prove that some subgroups of the group DiffF (M) are topological groups with the F-compact-open topology. Throughout this paper, we use smoothness of the class C∞. [ABSTRACT FROM AUTHOR]

This survey is devoted to particular problems of extrinsic geometry of foliations, which, roughly speaking, describes how leaves (or single submanifolds) are located in the ambient pseudo- Riemannian space. We discuss the following topics with the mixed scalar curvature: integral formulas and splitting of foliations, prescribing the mixed curvature of foliations, and variations of functionals defined on foliations, which seem to be central in extrinsic geometry. [ABSTRACT FROM AUTHOR]

The aim of this paper is to classify codimension 1 foliations F$\operatorname{\mathcal {F}}$ with canonical singularities and ν(KF)<3$\nu (K_{\operatorname{\mathcal {F}}}) < 3$ on threefolds of general type. I prove a classification result for foliations satisfying these conditions and having nontrivial algebraic part. We also describe purely transcendental foliations F$\operatorname{\mathcal {F}}$ with the canonical class KF$K_{\operatorname{\mathcal {F}}}$ being not big on manifolds of general type in any dimension, assuming that F$\operatorname{\mathcal {F}}$ is nonsingular in codimension 2. [ABSTRACT FROM AUTHOR]

We construct infinite countable families of pairwise topologically nonconjugate free chaotic groups of homeomorphisms on closed topological manifolds of different dimension. We find new invariants of suspended foliations in some complete subcategory of the category of foliations and use these results to construct infinite countable families of pairwise nonisomorphic chaotic topological foliations of an arbitrary even codimension on closed manifolds. [ABSTRACT FROM AUTHOR]

Let T = S 1 × D 2 be the solid torus, F the Morse–Bott foliation on T into 2-tori parallel to the boundary and one singular circle S 1 × 0 , which is the central circle of the torus T, and D (F , ∂ T) the group of diffeomorphisms of T fixed on ∂ T and leaving each leaf of the foliation F invariant. We prove that D (F , ∂ T) is contractible. Gluing two copies of T by some diffeomorphism between their boundaries, we will get a lens space L p , q with a Morse–Bott foliation F p , q obtained from F on each copy of T. We also compute the homotopy type of the group D (F p , q) of diffeomorphisms of L p , q leaving invariant each leaf of F p , q . [ABSTRACT FROM AUTHOR]

We give some properties of semi-symmetric pseudo-Riemannian manifolds as an indecomposable irreducible Ricci pseudo-Riemannian manifold (i.e. the minimal polynomial of its Ricci operator is irreducible) is semi symmetric if and only if it is locally symmetric. We also show that any semi-symmetric pseudo-Riemannian manifold will be foliated. Moreover, if the metric is Lorentzian, the Ricci operator has only real eigenvalues and more precisely, on each leaf, it is diagonalizable with at most a single non zero eigenvalue or isotropic. [ABSTRACT FROM AUTHOR]

We introduce the vertex algebra cohomology of foliations on complex curves. Generalizing the classical case, the orthogonality condition with respect to a product of elements of the double complexes associated to a grading-restricted vertex algebra matrix elements leads to the construction of cohomology invariants of codimension one foliations. [ABSTRACT FROM AUTHOR]

We revisit the interplay between the mass, the center of mass and the large scale behavior of certain isoperimetric quotients in the setting of asymptotically flat 3-manifolds (both without and with a non-compact boundary). In the boundaryless case, we first check that the isoperimetric deficits involving the total mean curvature recover the ADM mass in the asymptotic limit, thus extending a classical result due to G. Huisken. Next, under a Schwarzschild asymptotics and assuming that the mass is positive we indicate how the implicit function method pioneered by R. Ye and refined by L.-H. Huang may be adapted to establish the existence of a foliation of a neighborhood of infinity satisfying the corresponding curvature conditions. Recovering the mass as the asymptotic limit of the corresponding relative isoperimetric deficit also holds true in the presence of a non-compact boundary, where we additionally obtain, again under a Schwarzschild asymptotics, a foliation at infinity by free boundary constant mean curvature hemispheres, which are shown to be the unique relative isoperimetric surfaces for all sufficiently large enclosed volume, thus extending to this setting a celebrated result by M. Eichmair and J. Metzger. Also, in each case treated here we relate the geometric center of the foliation to the center of mass of the manifold as defined by Hamiltonian methods. [ABSTRACT FROM AUTHOR]

We prove that a generic holomorphic foliation on a weighted projective plane has no algebraic solutions when the degree is big enough. We also prove an analogous result for foliations on Hirzebruch surfaces. [ABSTRACT FROM AUTHOR]

Given a holomorphic singular foliation |${\mathcal {F}}$| of |$({\mathbb {C}}^n,0)$|⁠ , we define |$\textrm {Iso}({\mathcal {F}})$| as the group of germs of biholomorphisms on |$({\mathbb {C}}^n,0)$| preserving |${\mathcal {F}}$|⁠ : |$\textrm {Iso}({\mathcal {F}})\!=\!\lbrace \Phi \in \textrm {Diff}({\mathbb {C}}^n,0)\,|\,\Phi ^*({\mathcal {F}})\!=\!{\mathcal {F}}\rbrace $|⁠. The normal subgroup of |$\textrm {Iso}({\mathcal {F}})$|⁠ , of biholomorphisms sending each leaf of |${\mathcal {F}}$| into itself, will be denoted as |$\textrm {Fix}({\mathcal {F}})$|⁠. The corresponding groups of formal biholomorphisms will be denoted as |$\widehat {\textrm {Iso}}({\mathcal {F}})$| and |$\widehat {\textrm {Fix}}({\mathcal {F}})$|⁠ , respectively. The purpose of this paper will be to study the quotients |$\textrm {Iso}({\mathcal {F}})/\textrm {Fix}({\mathcal {F}})$| and |$\widehat {\textrm {Iso}}({\mathcal {F}})/\widehat {\textrm {Fix}}({\mathcal {F}})$|⁠ , mainly in the case of codimension one foliation. [ABSTRACT FROM AUTHOR]

We study foliations by curves on the three-dimensional projective space with no isolated singularities, which is equivalent to assuming that the conormal sheaf is locally free. We provide a classification of the topological and algebraic invariants of the conormal sheaves and singular schemes for such foliations by curves, up to degree 3. In particular, we prove that foliations by curves of degree 1 or 2 are contained in a pencil of planes or are Legendrian, and are given by the complete intersection of two codimension one distributions. Furthermore, we prove that the conormal sheaf of a foliation by curves of degree 3 with reduced singular scheme either splits as a sum of line bundles or is an instanton bundle. For degree larger than 3, we focus on two classes of foliations by curves, namely Legendrian foliations and those whose conormal sheaf is a twisted null-correlation bundle. We give characterizations of such foliations, describe their singular schemes and their moduli spaces. [ABSTRACT FROM AUTHOR]

We give a classification of the dual graphs of the exceptional divisors on the minimal resolutions of log canonical foliation singularities on surfaces. As an application, we show the set of foliated minimal log discrepancies for foliated surface triples satisfies the ascending chain condition and a Grauert–Riemenschneider–type vanishing theorem for foliated surfaces with special log canonical foliation singularities. [ABSTRACT FROM AUTHOR]

We prove that a logarithmic foliation corresponding to a generic line arrangement of |$d+1 \geq 3$| lines in the complex plane, with pairwise natural and co-prime residues, is a smooth point of the center set of plane foliations (vector fields) of degree |$d$|⁠. [ABSTRACT FROM AUTHOR]

Let (M , ω) be a compact symplectic manifold. We denote by ω the space of all almost complex structure compatible with ω. ω has a natural foliation structure with the complexified orbit as leaf. We obtain an explicit formula of the Hessian of Hermitian Calabi functional at an extremal almost Kähler metric in ω . We prove that the Hessian of Hermitian Calabi functional is semi-positive definite at critical point when restricted to a complexified orbit, as corollaries we obtain some results analogy to Kähler case. We also show weak parabolicity of the Hermitian Calabi flow. [ABSTRACT FROM AUTHOR]

We study the behavior of Sobolev functions and mappings on the Carnot groups with the left invariant sub-Riemannian metric. We obtain some sufficient conditions for a Sobolev function to be locally Hölder continuous (in the Carnot–Carathéodory metric) on almost every hypersurface of a given foliation. As an application of these results we show that a quasimonotone contact mapping of class of Carnot groups is continuous, -differentiable almost everywhere, and has the -Luzin property. [ABSTRACT FROM AUTHOR]

In this paper we study transversely holomorphic foliations of complex codimension one with a transversely homogeneous complex transverse structure. We prove that the only cases are the transversely additive, affine and projective cases. We shall focus on the transversely affine case and describe the holonomy of a leaf which is “at the infinity” with respect to this structure and prove this is a solvable group. Using this we are able to prove linearization results for the foliation under the assumption of existence of some hyperbolic map in the holonomy group. Such foliations will then be given by simple-poles closed transversely meromorphic one-forms. [ABSTRACT FROM AUTHOR]

In this short note, we prove two properties of symplectic pairs on a four-manifold: firstly we prove that two transversal orientable foliations of codimension two, which are taut for the same Riemannian metric, are the characteristic foliations of a symplectic pair; secondly, we characterize intrinsically harmonic 2-forms of rank two as part of a symplectic pair. [ABSTRACT FROM AUTHOR]

Manifolds with boundary, with corners, b -manifolds and foliations model configuration spaces for particles moving under constraints and can be described as E -manifolds. E -manifolds were introduced in Nest and Tsygan (2001 Asian J. Math. 5 599–635) and investigated in depth in Miranda and Scott (2021 Rev. Mat. Iberoam. 37 1207–24). In this article we explore their physical facets by extending gauge theories to the E -category. Singularities in the configuration space of a classical particle can be described in several new scenarios unveiling their Hamiltonian aspects on an E -symplectic manifold. Following the scheme inaugurated in Weinstein (1978 Lett. Math. Phys. 2 417–20), we show the existence of a universal model for a particle interacting with an E -gauge field. In addition, we generalise the description of phase spaces in Yang–Mills theory as Poisson manifolds and their minimal coupling procedure, as shown in Montgomery (1986 PhD Thesis University of California, Berkeley), for base manifolds endowed with an E -structure. In particular, the reduction at coadjoint orbits and the shifting trick are extended to this framework. We show that Wong's equations, which describe the interaction of a particle with a Yang–Mills field, become Hamiltonian in the E -setting. We formulate the electromagnetic gauge in a Minkowski space relating it to the proper time foliation and we see that our main theorem describes the minimal coupling in physical models such as the compactified black hole. [ABSTRACT FROM AUTHOR]

Consider the Hamiltonian action of a torus on a transversely symplectic foliation that is also Riemannian. When the transverse hard Lefschetz property is satisfied, we establish a foliated version of the Kirwan injectivity theorem and use it to study Hamiltonian torus actions on transversely Kähler foliations. Among other things, we prove a foliated analogue of the Carrell–Liberman theorem. As an application, this confirms a conjecture raised by Battaglia–Zaffran on the basic Hodge numbers of symplectic toric quasifolds. Our methods also allow us to present a symplectic approach to the calculation of the basic Betti numbers of symplectic toric quasifolds. [ABSTRACT FROM AUTHOR]

We formalize the concepts of holomorphic affine and projective structures along the leaves of holomorphic foliations by curves on complex manifolds. We show that many foliations admit such structures, we provide local normal forms for them at singular points of the foliation, and we prove some index formulae in the case where the ambient manifold is compact. As a consequence of these, we establish that a regular foliation of general type on a compact algebraic manifold of even dimension does not admit a foliated projective structure. Finally, we classify foliated affine and projective structures along regular foliations on compact complex surfaces. [ABSTRACT FROM AUTHOR]

In this survey paper the ultrahyperbolic equation in dimension four is discussed from a geometric, analytic and topological point of view. The geometry centres on the canonical neutral metric on the space of oriented geodesics of 3-dimensional space-forms, the analysis discusses a mean value theorem for solutions of the equation and presents a new solution of the Cauchy problem over a certain family of null hypersurfaces, while the topology relates to generalizations of codimension two foliations of 4-manifolds. [ABSTRACT FROM AUTHOR]

A weak f-contact structure, introduced in our recent works, generalizes the classical f-contact structure on a smooth manifold, and its characteristic distribution defines a totally geodesic foliation with flat leaves. We find the splitting tensor of this foliation and use it to show positive definiteness of the Jacobi operators in the characteristic directions and to obtain a topological obstruction (including the Adams number) to the existence of weak f-K-contact manifolds, and prove integral formulas for a compact weak f-contact manifold. Based on applications of the weak f-contact structure in Riemannian contact geometry considered in the article, we expect that this structure will also be fruitful in theoretical physics, e.g., in QFT. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a sub-family of the usual generalized Wronskians, which we call geometric generalized Wronskians. It is well known that one can test linear dependance of holomorphic functions (of several variables) via the identical vanishing of generalized Wronskians. We show that such a statement remains valid if one tests the identical vanishing only on geometric generalized Wronskians. It turns out that geometric generalized Wronskians allow to define intrinsic objects on projective varieties polarized with an ample line bundle: in this setting, the lack of existence of global functions is compensated by global sections of powers of the fixed ample line bundle. Geometric generalized Wronskians are precisely defined so that their local evaluations on such global sections globalize up to a positive twist by the ample line bundle. We then give three applications of the construction of geometric generalized Wronskians: one in intermediate hyperbolicity and two in foliation theory. In intermediate hyperbolicity, we show the algebraic degeneracy of holomorphic maps from |${\mathbb {C}}^{p}$| to a Fermat hypersurface in |$\textbf {P}^{N}$| of degree |$\delta> (N+1)(N-p)$|⁠ : this interpolates between two well-known results, namely for |$p=1$| (first proved via Nevanlinna theory) and |$p=N-1$| (in which case the Fermat hypersurface is of general type). The first application in foliation theory provides a criterion for algebraic integrability of leaves of foliations: our criterion is not optimal in view of current knowledge, but has the advantage of having an elementary proof. Our second application deals with positivity properties of adjoint line bundles of the form |$K_{\mathcal {F}} + L $|⁠ , where |$K_{\mathcal {F}}$| is the canonical bundle of a regular foliation |$\mathcal {F}$| on a smooth projective variety |$X$|⁠ , and where |$L$| is an ample line bundle on |$X$|⁠. [ABSTRACT FROM AUTHOR]