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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 consider the horospherical foliation on any invariant subvariety in the moduli space of translation surfaces. This foliation can be described dynamically as the strong unstable foliation for the geodesic flow on the invariant subvariety, and geometrically, it is induced by the canonical splitting of C -valued cohomology into its real and imaginary parts. We define a natural volume form on the leaves of this foliation, and define horospherical measures as those measures whose conditional measures on leaves are given by the volume form. We show that the natural measures on invariant subvarieties, and in particular, the Masur-Veech measures on strata, are horospherical. We show that these measures are the unique horospherical measures giving zero mass to the set of surfaces with horizontal saddle connections, extending work of Lindenstrauss-Mirzakhani and Hamenstädt for principal strata. We describe all the leaf closures for the horospherical foliation. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study codimension one foliations on rational homogeneous spaces, with a focus on the moduli space of foliations of low degree on Grassmannians and cominuscule spaces. Using equivariant techniques, we show that codimension one degree zero foliations on (ordinary, orthogonal, symplectic) Grassmannians of lines, some spinor varieties, some Lagrangian Grassmannians, the Cayley plane (an E 6 -variety) and the Freudenthal variety (an E 7 -variety) are identified with restrictions of foliations on the ambient projective space. We also provide some evidence that such results can be extended beyond these cases. [ABSTRACT FROM AUTHOR]

Abstract In this paper we define new K-theoretic secondary invariants attached to a Lie groupoid G. The receptacle for these invariants is the K-theory of C r ⁎ (G a d ∘) (where G a d ∘ is the adiabatic deformation G restricted to the interval [ 0 , 1)). Our construction directly generalises the cases treated in [29,30] , in the setting of the Coarse Geometry, to more involved geometrical situations, such as foliations. Moreover we tackle the problem of producing a wrong-way functoriality between adiabatic deformation groupoid K-groups associated to transverse maps. This extends the construction of the lower shriek map in [6]. Furthermore we prove a Lie groupoid version of the Delocalized APS Index Theorem of Piazza and Schick. Finally we give a product formula for secondary invariants. [ABSTRACT FROM AUTHOR]

Monotone recurrence relations give rise to a class of dynamical systems on the high-dimensional cylinder which generalizes the class of monotone twist maps. A solution of a monotone recurrence relation corresponds to an equilibrium of the generalized Frenkel–Kontorova (FK) model. The Aubry–Mather theory for monotone recurrence relations says that for each ω ∈ R there is a Birkhoff minimizer with rotation number ω . Whether all Birkhoff minimizers of rotation number ω form a foliation is a question like whether there is an invariant circle with rotation number ω for a monotone twist map. In this paper we give a criterion for the existence of minimal foliations and study its continuity. The depinning force F d ( ω ) , depending on rotation numbers, is a critical value of the driving force for the FK model, under which there are Birkhoff equilibria and hence the system is pinned, and above which there are no Birkhoff equilibria of rotation number ω and the system is sliding. We show that all Birkhoff minimizers with irrational rotation number ω form a foliation if and only if F d ( ω ) = 0 . If ω = p / q is rational, then F d ( p / q ) = 0 if and only if all ( p , q ) -periodic Birkhoff minimizers constitute a foliation. Moreover, we show that F d ( ω ) is continuous at irrational ω and Hölder continuous at Diophantine points, and it also depends continuously on system parameters. [ABSTRACT FROM AUTHOR]

In this paper we study the algebraic ranks of foliations on Q -factorial normal projective varieties. We start by establishing a Kobayashi-Ochiai's theorem for Fano foliations in terms of algebraic rank. We then investigate the local positivity of the anti-canonical divisors of foliations, obtaining a lower bound for the algebraic rank of a foliation in terms of Seshadri constant. We describe those foliations whose algebraic rank slightly exceeds this bound and classify Fano foliations on smooth projective varieties attaining this bound. Finally we construct several examples to illustrate the general situation, which in particular allow us to answer a question asked by Araujo and Druel on the generalised indices of foliations. [ABSTRACT FROM AUTHOR]

In this work we prove a Baum–Bott type residue theorem for flags of holomorphic foliations. We prove some relations between the residues of the flag and the residues of their correspondent foliations. We define the Nash residue for flags and we give a partial answer to the Baum–Bott type rationality conjecture in this context. [ABSTRACT FROM AUTHOR]

Given a closed manifold M and a vector bundle ξ of rank n over M , by gluing two copies of the disc bundle of ξ , we can obtain a closed manifold D ( ξ , M ) , the so-called double manifold. In this paper, we firstly prove that each sphere bundle S r ( ξ ) of radius r > 0 is an isoparametric hypersurface in the total space of ξ equipped with a connection metric, and for r > 0 small enough, the induced metric of S r ( ξ ) has positive Ricci curvature under the additional assumptions that M has a metric with positive Ricci curvature and n ≥ 3 . As an application, if M admits a metric with positive Ricci curvature and n ≥ 2 , then we construct a metric with positive Ricci curvature on D ( ξ , M ) . Moreover, under the same metric, D ( ξ , M ) admits a natural isoparametric foliation. For a compact minimal isoparametric hypersurface Y n in S n + 1 ( 1 ) , which separates S n + 1 ( 1 ) into S + n + 1 and S − n + 1 , one can get double manifolds D ( S + n + 1 ) and D ( S − n + 1 ) . Inspired by Tang, Xie and Yan's work on scalar curvature of such manifolds with isoparametric foliations (cf. [25] ), we study Ricci curvature of them with isoparametric foliations in the last part. [ABSTRACT FROM AUTHOR]

We show that the properties of admitting a co-oriented taut foliation and having a left-orderable fundamental group are equivalent for rational homology 3-sphere graph manifolds and relate them to the property of not being a Heegaard–Floer L-space. This is accomplished in several steps. First we show how to detect families of slopes on the boundary of a Seifert fibred manifold in four different fashions—using representations, using left-orders, using foliations, and using Heegaard–Floer homology. Then we show that each method of detection determines the same family of detected slopes. Next we provide necessary and sufficient conditions for the existence of a co-oriented taut foliation on a graph manifold rational homology 3-sphere, respectively a left-order on its fundamental group, which depend solely on families of detected slopes on the boundaries of its pieces. The fact that Heegaard–Floer methods can be used to detect families of slopes on the boundary of a Seifert fibred manifold combines with certain conjectures in the literature to suggest an L-space gluing theorem for rational homology 3-sphere graph manifolds as well as other interesting problems in Heegaard–Floer theory. [ABSTRACT FROM AUTHOR]

We prove the strong Weinstein conjecture for closed contact manifolds that appear as the concave boundary of a symplectic cobordism admitting an essential local foliation by holomorphic spheres. [ABSTRACT FROM AUTHOR]

It is known that a spectral gap condition is required for the existence and smoothness of invariant manifolds and invariant foliations near a fixed point. However, any planar mapping with a unipotent linear part, having a center manifold on the whole phase plane, does not have such a gap. In this paper, we give the existence and smoothness of invariant manifolds, which are invariant submanifolds of the center manifold, for the aforementioned planar mapping. Corresponding to the invariant submanifolds, we further prove the existence and smoothness of invariant foliations on the center manifold. [ABSTRACT FROM AUTHOR]

Let (M , g T M) be a noncompact (not necessarily complete) enlargeable Riemannian manifold in the sense of Gromov-Lawson and F an integrable subbundle of TM. Let k F be the leafwise scalar curvature associated to g F = g T M | F. We show that if either TM or F is spin, then inf (k F) ≤ 0. This generalizes the famous result of Gromov-Lawson on enlargeable manifolds to the case of foliations. It also extends an ansatz of Gromov on hyper-Euclidean spaces to general enlargeable Riemannian manifolds, as well as recent results on compact enlargeable foliated manifolds due to Benameur-Heitsch et al. to the noncompact situation. [ABSTRACT FROM AUTHOR]

We consider holomorphic maps defined in an annulus around R / Z in C / Z. E. Risler proved that in a generic analytic family of such maps f ζ that contains a Brjuno rotation f 0 (z) = z + α , all maps that are conjugate to this rotation form a codimension-1 analytic submanifold near f 0. In this paper, we obtain the Risler's result as a corollary of the following construction. We introduce a renormalization operator on the space of univalent maps in a neighborhood of R / Z. We prove that this operator is hyperbolic, with one unstable direction corresponding to translations. We further use a holomorphic motions argument and Yoccoz's theorem to show that its stable foliation consists of diffeomorphisms that are conjugate to rotations. [ABSTRACT FROM AUTHOR]

We prove an analogue of the Hitchin-Kobayashi correspondence for compact, oriented, taut Riemannian foliated manifolds with transverse Hermitian structure. In particular, our Hitchin-Kobayashi theorem holds on any compact Sasakian manifold. We define the notion of stability for foliated Hermitian vector bundles with transverse holomorphic structure and prove that such bundles admit a basic Hermitian-Einstein connection if and only if they are polystable. Our proof is obtained by adapting the proof by Uhlenbeck and Yau to the foliated setting. We relate the transverse Hermitian-Einstein equations to higher dimensional instanton equations and in particular we look at the relation to higher contact instantons on Sasakian manifolds. For foliations of complex codimension 1, we obtain a transverse Narasimhan-Seshadri theorem. We also demonstrate that the weak Uhlenbeck compactness theorem fails in general for basic connections on a foliated bundle. This shows that not every result in gauge theory carries over to the foliated setting. [ABSTRACT FROM AUTHOR]

We develop the deformation theory of symplectic foliations, i.e. regular foliations equipped with a leafwise symplectic form. The main result of this paper is that each symplectic foliation has an attached L ∞ -algebra controlling its deformation problem. Indeed, viewing symplectic foliations as regular Poisson structures, we establish a one-to-one correspondence between the small deformations of a given symplectic foliation and the Maurer-Cartan elements of the associated L ∞ -algebra. Using this, we show that infinitesimal deformations of symplectic foliations can be obstructed. Further, we relate symplectic foliations with foliations on one side and with (arbitrary) Poisson structures on the other, showing that obstructed infinitesimal deformations of the former may give rise to unobstructed deformations of the latter. [ABSTRACT FROM AUTHOR]

In this paper we consider the uniqueness problem of the constant mean curvature spheres in asymptotically flat 3-manifolds. We require the metric to have the form g i j = δ i j + h i j with h i j = O 4 ( r − 1 ) and R = O ( r − 3 − τ ) , τ > 0 . We do not require the metric to be close to Schwarzschild metric in any sense or to satisfy RT conditions. We prove that, when the mass is not 0, stable CMC spheres that separate a certain compact part from the infinity satisfy the radius pinching estimate r 1 ≤ C r 0 , which in many cases is critical to prove the uniqueness of the CMC spheres. As applications of this estimate, we remove the radius conditions of the uniqueness result in [8] and [15] in some special cases. [ABSTRACT FROM AUTHOR]

The focal sets of isoparametric hypersurfaces in spheres with g = 4 are all Willmore submanifolds, being minimal but mostly non-Einstein [25,22] . Inspired by A. Gray's view, the present paper shows that these focal sets are all A -manifolds but rarely Ricci parallel, except possibly for the only unclassified case. As a byproduct, it gives infinitely many simply-connected examples to the problem 16.56(i) of Besse concerning generalizations of the Einstein condition. [ABSTRACT FROM AUTHOR]

We classify the polynomial vector fields X on C 2 which have simply-connected trajectories and satisfy d P ( X ) = 1 for P in C [ x , y ] . In particular, we obtain that such vector fields are complete and have all its trajectories proper and of type C . Finally, we apply this result to classify surjective C -derivations of C [ x , y ] , and provide a proof of a conjecture of Cerveau [7] . [ABSTRACT FROM AUTHOR]

We prove generic semipositivity of the tangent bundle of a non-uniruled Calabi–Yau variety in positive characteristic. We also construct an example of a nef line bundle in characteristic zero, whose each reduction to positive characteristic is not nef. [ABSTRACT FROM AUTHOR]

Given a non-compact, simply connected homogeneous three-manifold X and a sequence { Ω n } n of isoperimetric domains in X with volumes tending to infinity, we prove that, as n → ∞ : (1) The radii of the Ω n tend to infinity. (2) The ratios Area ( ∂ Ω n ) Vol ( Ω n ) converge to the Cheeger constant Ch ( X ) , which we also prove to be equal to 2 H ( X ) where H ( X ) is the critical mean curvature of X . (3) The values of the constant mean curvatures H n of the boundary surfaces ∂ Ω n converge to 1 2 Ch ( X ) . Furthermore, when Ch ( X ) is positive, we prove that for n large, ∂ Ω n is well-approximated in a natural sense by the leaves of a certain foliation of X , where every leaf of the foliation is a surface of constant mean curvature H ( X ) . [ABSTRACT FROM AUTHOR]

We express Witten's deformation of Morse functions using deformation to the normal cone and C ⁎ -modules. This allows us to obtain asymptotics of the 'large eigenvalues'. This is then applied to the case of Morse-Bott functions inspired by [2]. Our methods extend to Morse functions along a foliation. We construct the Witten deformation using any generic function on an arbitrary foliation on a compact manifold and establish the compactness of its resolvent. When the foliation has a holonomy invariant transverse measure we show that our result implies Morse inequalities obtained by Connes and Fack [10] in a slightly more general situation. [ABSTRACT FROM AUTHOR]

Abstract: In order to understand the linearization problem around a leaf of a singular foliation, we extend the familiar holonomy map from the case of regular foliations to the case of singular foliations. To this aim we introduce the notion of holonomy transformation. Unlike the regular case, holonomy transformations cannot be attached to classes of paths in the foliation, but rather to elements of the holonomy groupoid of the singular foliation. [Copyright &y& Elsevier]

Abstract: In this paper we address Fano foliations on complex projective varieties. These are foliations whose anti-canonical class is ample. We focus our attention on a special class of Fano foliations, namely del Pezzo foliations on complex projective manifolds. We show that these foliations are algebraically integrable, with one exceptional case when the ambient space is . We also provide a classification of del Pezzo foliations with mild singularities. [Copyright &y& Elsevier]

Abstract: We study local G-shtukas with level structure over a base scheme whose Newton polygons are constant on the base. We show that after a finite base change and after passing to an étale covering, such a local G-shtuka is isogenous to a completely slope divisible one, generalizing corresponding results for p-divisible groups by Oort and Zink. As an application we establish a product structure up to finite surjective morphism on the closed Newton stratum of the universal deformation of a local G-shtuka, similarly to Oortʼs foliations for p-divisible groups and abelian varieties. This also yields bounds on the dimensions of affine Deligne–Lusztig varieties and proves equidimensionality of affine Deligne–Lusztig varieties in the affine Grassmannian. [Copyright &y& Elsevier]

Abstract: We refine the cyclic cohomological apparatus for computing the Hopf cyclic cohomology of the Hopf algebras associated to infinite primitive Cartan–Lie pseudogroups, and for the transfer of their characteristic classes to foliations. The main novel feature is the precise identification as a Hopf cyclic complex of the image of the canonical homomorphism from the Gelfand–Fuks complex to the Bott complex for equivariant cohomology. This provides a convenient new model for the Hopf cyclic cohomology of the geometric Hopf algebras, which allows for an efficient transport of the Hopf cyclic classes via characteristic homomorphisms. We illustrate the latter aspect by indicating how to realize the universal Hopf cyclic Chern classes in terms of explicit cocycles in the cyclic cohomology of étale foliation groupoids. [Copyright &y& Elsevier]

Abstract: For a Lie groupoid with a twisting σ (a -principal bundle over ), we use the (geometric) deformation quantization techniques supplied by Connes tangent groupoids to define an analytic index morphism in twisted K-theory. In the case the twisting is trivial we recover the analytic index morphism of the groupoid. Display Omitted For a smooth foliated manifold with twistings on the holonomy groupoid we prove the twisted analog of the Connes–Skandalis longitudinal index theorem. When the foliation is given by fibers of a fibration, our index coincides with the one recently introduced by Mathai, Melrose, and Singer. We construct the pushforward map in twisted K-theory associated to any smooth (generalized) map and a twisting σ on the holonomy groupoid , next we use the longitudinal index theorem to prove the functoriality of this construction. We generalize in this way the wrong way functoriality results of Connes and Skandalis when the twisting is trivial and of Carey and Wang for manifolds. [Copyright &y& Elsevier]

Abstract: We construct fiber-preserving anti-symplectic involutions for a large class of symplectic manifolds with Lagrangian torus fibrations. In particular, we treat the K3 surface and the six-dimensional examples constructed by Castaño-Bernard and Matessi (2009) , which include a six-dimensional symplectic manifold homeomorphic to the quintic threefold. We interpret our results as corroboration of the view that in homological mirror symmetry, an anti-symplectic involution is the mirror of duality. In the same setting, we construct fiber-preserving symplectomorphisms that can be interpreted as the mirror to twisting by a holomorphic line bundle. [ABSTRACT FROM AUTHOR]

Abstract: We discuss the Siciak–Zaharjuta extremal function of a real convex body in , a solution of the homogeneous complex Monge–Ampère equation on the exterior of the convex body. We determine several conditions under which a foliation by holomorphic curves can be found in the complement of the convex body along which the extremal function is harmonic. We study a variational problem for holomorphic disks in projective space passing through prescribed points at infinity. The extremal curves are all complex quadratic curves, and the geometry of such curves allows for the determination of the leaves of the foliation by simple geometric criteria. As a byproduct we encounter a new invariant of an exterior domain, the Robin indicatrix, which is in some cases the dual of the Kobayashi indicatrix for a bounded domain. Finally, we construct extremal curves for two non-convex bodies in . [Copyright &y& Elsevier]

The (measure-theoretical) entropy of a diffeomorphism along an expanding invariant foliation is the rate of complexity generated by the diffeomorphism along the leaves of the foliation. We prove that this number varies upper semi-continuously with the diffeomorphism (C 1 topology), the invariant measure (weak* topology) and the foliation itself in a suitable sense. This has several important consequences. For one thing, it implies that the set of Gibbs u -states of C 1 + partially hyperbolic diffeomorphisms is an upper semi-continuous function of the map in the C 1 topology. Another consequence is that the sets of partially hyperbolic diffeomorphisms with mostly contracting or mostly expanding center are C 1 open. New examples of partially hyperbolic diffeomorphisms with mostly expanding center are provided, and the existence of physical measures for C 1 residual subset of diffeomorphisms is discussed. We also provide a new class of robustly transitive diffeomorphisms: every C 2 volume preserving, accessible partially hyperbolic diffeomorphism with one dimensional center and non-vanishing center exponent is C 1 robustly transitive (among neighborhood of diffeomorphisms which are not necessarily volume preserving). [ABSTRACT FROM AUTHOR]

Abstract: We study completely reducible fibers of pencils of hypersurfaces on and associated codimension one foliations of . Using methods from theory of foliations we obtain certain upper bounds for the number of these fibers as functions only of n. Equivalently this gives upper bounds for the dimensions of resonance varieties of hyperplane arrangements. We obtain similar bounds for the dimensions of the characteristic varieties of the arrangement complements. [Copyright &y& Elsevier]

We show that for any locally compact Hausdorff space Y with finite covering dimension and for any continuous flow R ↷ Y , the resulting crossed product C ⁎ -algebra C 0 (Y) ⋊ R has finite nuclear dimension. This generalizes previous results for free flows, where this was proved using Rokhlin dimension techniques. As an application, we obtain bounds for the nuclear dimension of C ⁎ -algebras associated to one-dimensional orientable foliations. This result is analogous to the one we obtained earlier for non-free actions of Z. Some novel techniques in our proof include the use of a conditional expectation constructed from the inclusion of a clopen subgroupoid, as well as the introduction of what we call fiberwise groupoid coverings that help us build a link between foliation C ⁎ -algebras and crossed products. [ABSTRACT FROM AUTHOR]

Abstract: We prove that homologically nontrivial generic smooth -parameter families of analytic discs in , , attached by their boundaries to a CR-manifold Ω, test CR-functions in the following sense: if a smooth function on Ω analytically extends into any analytic discs from the family, then the function satisfies tangential CR-equations on Ω. In particular, we give an answer (Theorem 1) to the following long standing open question, so called strip-problem, earlier solved only for special families (mainly for circles): given a smooth one-parameter family of Jordan curves in the plane and a function f admitting holomorphic extension inside each curve, must f be holomorphic on the union of the curves? We prove, for real-analytic functions and arbitrary generic real-analytic families of curves, that the answer is “yes,” if no point is surrounded by all curves from the family. The latter condition is essential. We generalize this result to characterization of complex curves in as real 2-manifolds admitting nontrivial families of attached analytic discs (Theorem 4). The main result implies fairly general Morera type characterization of CR-functions on hypersurfaces in in terms of holomorphic extensions into three-parameter families of attached analytic discs (Theorem 2). One of the applications is confirming, in real-analytic category, the Globevnik–Stout conjecture (Theorem 3) on boundary values of holomorphic functions. It is proved that a smooth function on the boundary of a smooth strictly convex domain in extends holomorphically inside the domain if it extends holomorphically into complex lines tangent to a given strictly convex subdomain. The proofs are based on a universal approach, namely, on the reduction to a problem of propagation, from the boundary to the interior, of degeneracy of CR-foliations of solid torus type manifolds (Theorem 2.2). [Copyright &y& Elsevier]

This paper presents an improved approach to the theory of harmonic measures for foliated spaces introduced by Garnett. This approach is based on a method for solving elliptic equations on foliated spaces and on the Hille–Yosida theory. The diffusion semigroup of a general Laplacian and its infinitesimal generator are made explicit. Applications of the path space to the dynamical study of a foliated space are described. In particular, the final section studies cocycles on foliated spaces, a formula for their asymptotic limit, and some analytic and geometric consequences. [Copyright &y& Elsevier]

On closed symplectically aspherical manifolds, by using Floer homology, Schwarz proved a classical result, i.e., that the action function of a nontrivial Hamiltonian diffeomorphism is not constant. In this article, we generalize Schwarz's theorem to the C 0 -case on closed aspherical surfaces. Our methods are purely topological and involve the theory of transverse foliations for dynamical systems of surfaces and the recent progress inspired by Le Calvez. As an application, we prove that the contractible fixed points set (and consequently the fixed points set) of a nontrivial Hamiltonian homeomorphism is not connected. We also get a similar result for an orientation preserving nonwandering point homeomorphism of the two-sphere. In the end, we give further applications based on the C 0 -Schwarz Theorem. [ABSTRACT FROM AUTHOR]

In this paper we obtain local rigidity results for linear Anosov diffeomorphisms in terms of Lyapunov exponents. More specifically, we show that given an irreducible linear hyperbolic automorphism L with simple real eigenvalues with distinct absolute values, any small perturbation preserving the volume and with the same Lyapunov exponents is smoothly conjugate to L. We also obtain rigidity results for skew products over Anosov diffeomorphisms. Given a volume preserving partially hyperbolic skew product diffeomorphism f 0 over an Anosov automorphism of the 2-torus, we show that for any volume preserving perturbation f of f 0 with the same average stable and unstable Lyapunov exponents, the center foliation is smooth. [ABSTRACT FROM AUTHOR]