arXiv research

Study on tautness tensor for Riemannian foliations.

Torus leaves play a crucial role in the theory of foliations. For example non-taut foliations admit a torus leaf (see the article of Goodman). In this paper, we study all the foliations near a torus leaf, and try to understand why sometimes it is taut, or non-taut (and Reebless). We focus on some crucial examples to un…

This paper concerns the problem of existence of taut foliations among 3-manifolds. Since the contribution of David Gabai, we know that closed 3-manifolds with non-trivial second homology group admit a taut foliations. The essential part of this paper focuses on Seifert fibered homology 3-spheres. The result is quite di…

Taut foliations map leaves to branched 2-sphere covers.

New foliations constructed from contact pairs, revealing flexible taut foliations.

Proof of contact structure from taut foliation for certain knots.

3-manifolds with Heegaard 2 admit taut foliations if their fundamental group is left-orderable.

We show the equivalence of several notions in the theory of taut foliations and the theory of tight contact structures. We prove equivalence, in certain cases, of existence of tight contact structures and taut foliations.

New findings on $L$-spaces and taut foliations in hyperbolic links.

Left orderability proven for certain 3-manifolds with specific foliations.

Floer homology detects taut foliations in rational homology spheres.

In this paper we present some new results on the tautness of Riemannian foliations in their historical context. The first part of the paper gives a short history of the problem. For a closed manifold, the tautness of a Riemannian foliation can be characterized cohomologically. We extend this cohomological characterizat…

We give an equivalent description of taut submanifolds of complete Riemannian manifolds as exactly those submanifolds whose normal exponential map has the property that every preimage of a point is a union of submanifolds. It turns out that every taut submanifold is also $\mathbb Z_2$-taut. We explicitely construct gen…

The Whitehead link exterior lacks most Euler class taut foliations.

This article proves that the parity of the number of Klein-bottle leaves in a smooth cooriented taut foliation is invariant under smooth deformations within taut foliations, provided that every Klein-bottle leaf involved in the counting has non-trivial linear holonomy.

This paper proves a converse to Thurston's 1976 observation for taut foliations on hyperbolic 3-manifolds.

We describe notions of tautness that arise in the study of $C^0$ foliations, $C^{1,0}$ or smoother foliations, and in geometry. We give examples to show that these notions are different, and discuss how these differences impact some classical foliation results. We construct examples of smoothly taut $C^{\infty,0}$ foli…

New foliations found in 3D spaces from positive braids.

Whitehead link surgeries are not L-spaces if they support taut foliations.

Let $M$ be a fibered 3-manifold with multiple boundary components. We show that the fiber structure of $M$ transforms to closely related transversely oriented taut foliations realizing all rational multislopes in some open neighborhood of the multislope of the fiber. Each such foliation extends to a taut foliation in t…

The paper constructs 3-manifolds with co-orientable taut foliations but no foliations with vanishing Euler class.

We construct taut foliations in every closed 3-manifold obtained by $r$-framed Dehn surgery along a positive 3-braid knot $K$ in $S^3$, where $r < 2g(K)-1$ and $g(K)$ denotes the Seifert genus of $K$. This confirms a prediction of the L-space Conjecture. For instance, we produce taut foliations in every non-L-space obt…

Study positive braid knots and their taut foliations, proving some L-space conjecture evidence.

We study the question of when cyclic branched covers of knots admit taut foliations, have left-orderable fundamental group, and are not L-spaces.

In this note we give a characterization of taut Riemannian foliations using the transverse divergence. This result turns out to be a convenient tool in the case of some standard examples. Furthermore, we show that a classical tautness result of Haefliger can be obtained in our particular setting as a straightforward co…

Sutured instanton Floer homology was introduced by Kronheimer and Mrowka. In this paper, we prove that for a taut balanced sutured manifold with vanishing second homology, the dimension of the sutured instanton Floer homology provides a bound on the minimal depth of all possible taut foliations on that balanced sutured…

We define a norm on the homology of a foliated manifold, which refines and majorizes the usual Gromov norm on homology. This norm depends in an upper semi-continuous way on the underlying foliation, in the geometric topology, and can therefore be used to study the question of which foliations arise as geometric limits …

New evidence supports the Euler class one conjecture for tight contact structures.

Constructs foliations from manifold group left orders.

The study proves knots and certain links support taut foliations.

For a riemannian foliation $\mathcal{F}$ on a closed manifold $M$, it is known that $\mathcal{F}$ is taut (i.e. the leaves are minimal submanifolds) if and only if the (tautness) class defined by the mean curvature form $κ_μ$ (relatively to a suitable riemannian metric $μ$) is zero. In the transversally orientable case…

Non-unimodular foliations have specific geometric properties.

New proof shows left orderability of 3-manifold groups with specific foliations.

We extend the Eliashberg-Thurston theorem on approximations of taut oriented $C^2$-foliations of 3-manifolds by both positive and negative contact structures to a large class of taut oriented $C^{1,0}$-foliations, where by $C^{1,0}$ foliation, we mean a foliation with continuous tangent plane field. These $C^{1,0}$-fol…

The paper studies Liouville structures for taut foliations and Anosov flows, proving their topological invariance.

We show that for any nontrivial knot in $S^3$, there is an open interval containing zero such that a Dehn surgery on any slope in this interval yields a 3-manifold with taut foliations. This generalizes a theorem of Gabai on zero frame surgery.

A taut foliation of a hyperbolic 3-manifold has the continuous extension property for leaves in almost every direction; that is, for each leaf of the universal cover of the foliation and almost every geodesic ray in the leaf, the limit of the ray in the universal cover of the 3-manifold is a well-defined point in the i…

In this article, we study the Euler class of taut foliations on the Dehn fillings of a $\mathbb{Q}$-homology solid torus. We give a necessary and sufficient condition for the Euler class of a foliation transverse to the core of the filling solid torus to vanish. We apply this condition to taut foliations on Dehn fillin…

Using deformations of foliations to contact structures as well as rigidity properties of Anosov foliations we provide infinite families of examples which show that the space of taut foliations in a given homotopy class of plane fields is in general not path connected. Similar methods also show that the space of represe…

This paper explores the conjecture that the following are equivalent for rational homology 3-spheres: having left-orderable fundamental group, having non-minimal Heegaard Floer homology, and admitting a co-orientable taut foliation. In particular, it adds further evidence in favor of this conjecture by studying these t…

A 3-manifold is foliar if it supports a codimension-one co-oriented taut foliation. Suppose $M$ is an oriented 3-manifold with connected boundary a torus, and suppose $M$ contains a properly embedded, compact, oriented, surface $R$ with a single boundary component that is Thurston norm minimizing in $H_2(M, \partial M)…

We discuss the decomposition of degree two basic cohomology for codimension four taut Riemannian foliation according to the holonomy invariant transversal almost complex structure J, and show that J is C pure and full. In addition, we obtain an estimate of the dimension of basic J-anti-invariant subgroup. These are the…

Whether every hyperbolic 3-manifold admits a tight contact structure or not is an open question. Many hyperbolic 3-manifolds contain taut foliations and taut foliations can be perturbed to tight contact structures. The first examples of hyperbolic 3-manifolds without taut foliations were constructed by Roberts, Sharesh…

We show that a graph manifold which is a Z-homology 3-sphere not homeomorphic to either the 3-sphere or the Poincaré homology 3-sphere admits a horizontal foliation. This combines with known results to show that the conditions of not being an L-space, of having a left-orderable fundamental group, and of admitting a co-…

If $Y$ is a closed orientable graph manifold, we show that $Y$ admits a coorientable taut foliation if and only if $Y$ is not an L-space. Combined with previous work of Boyer and Clay, this implies that $Y$ is an L-space if and only if $π_1(Y)$ is not left-orderable.

Torus decomposition shows foliation detected slopes for glued knot manifolds.

We study the left-orderability of the fundamental groups of cyclic branched covers of links which admit co-oriented taut foliations. In particular we do this for cyclic branched covers of fibred knots in integer homology $3$-spheres and cyclic branched covers of closed braids. The latter allows us to complete the proof…

Study shows universal circles for taut foliations in 3-manifolds.