Study co-dimension one area-minimizing currents with tangentially immersed boundaries and co-oriented mean curvature.
problem Understanding area-minimizing currents with specific boundary conditions.
method Introducing and studying co-dimension one area-minimizing currents with tangentially immersed boundaries and co-oriented mean curvature.
result Any such currents are supported in a smooth hypersurface near the boundary, with tangent cones being hyperplanes of constant orientation but non-constant multiplicity.
We prove several topological properties of linear Weingarten surfaces of Bryant type, as wave fronts in hyperbolic 3-space. For example, we show the orientability of such surfaces, and also co-orientability when they are not flat. Moreover, we show an explicit formula of the non-holomorphic hyperbolic Gauss map via ano…
We study compatible toric Sasaki metrics with constant scalar curvature on co-oriented compact toric contact manifolds of Reeb type of dimension at least 5. These metrics come in rays of transversal homothety due to the possible rescaling of the Reeb vector fields. We prove that there exist Reeb vector fields for which…
We show that there exist infinitely many pairwise distinct non-closed G_2-manifolds (some of which have holonomy full G_2) such that they admit co-oriented contact structures and have co-oriented contact submanifolds which are also associative. Along the way, we prove that there exists a tubular neighborhood N of every…
Study area-minimizing currents with specific boundary properties.
problem Understanding the structure of area-minimizing currents with tangentially immersed boundaries.
method Analyzes n-dimensional area-minimizing currents with boundary constraints. result Near the boundary of the current, the structure is either uncontrolled or a union of controlled hypersurfaces.
Geometric cohomology model uses co-oriented maps to define a product structure.
problem Constructing a geometric model for cohomology of smooth manifolds.
method Develops a cochain complex model based on co-oriented smooth maps, focusing on their pull-back product structure.
result Geometric cochains with a partially defined product structure induce the cup product in cohomology.
An affine hypersurface (AH) structure is a pair comprising a conformal structure and a projective structure such that for any torsion-free connection representing the projective structure the completely trace-free part of the covariant derivative of any metric representing the conformal structure is completely symmetri…
The paper constructs 3-manifolds with co-orientable taut foliations but no foliations with vanishing Euler class.
problem Constructing 3-manifolds with co-orientable taut foliations but no foliations with vanishing Euler class.
method Constructs infinitely many rational homology 3-spheres using Dehn surgeries and Heegaard Floer homology.
result Found rational homology 3-spheres that admit co-orientable taut foliations but none with vanishing Euler class.
The study shows how to create co-orientable taut foliations in specific Dehn fillings.
problem Creating co-orientable taut foliations in Dehn fillings of pseudo-Anosov mapping tori.
method Analyzing the stable foliation and using Dehn filling with specific multislopes.
result Dehn fillings of pseudo-Anosov mapping tori with co-orientation-reversing monodromy admit co-orientable taut foliations.
Rectangular diagrams help analyze foliations in 3-sphere.
problem Analyzing foliations in 3-sphere complements.
method Introduced rectangular diagrams for foliations and links.
result Any co-orientable finite depth foliation can be presented by a compatible rectangular diagram.
In this article I propose a new method for reducing a co-oriented contact manifold M equipped with an action of a Lie group G by contact transformations. With a certain regularity and integrality assumption the contact quotient Mμ at $μ\in \fg^*$ is a naturally a co-oriented contact orbifold which is independent of …
New proof shows left orderability of 3-manifold groups with specific foliations.
problem Left orderability of 3-manifold groups with co-orientable taut foliations.
method Analyzes co-orientable taut foliations with orderable cataclysm to prove left orderability of 3-manifold fundamental groups.
result Proves left orderability of 3-manifold groups with specific foliations.
For Γ1-structures on 3-manifolds, we give a very simple proof of Thurston's regularization theorem, first proved in \cite{thurston}, without using Mather's homology equivalence. Moreover, in the co-orientable case, the resulting foliation can be chosen of a precise kind, namely an "open book foliation modified by su…
Study hypoelliptic operators on Carnot manifolds, extending index theory results.
problem Index theory of hypoelliptic operators on Carnot manifolds.
method Operator K-theory and geometric K-homology.
result Compute Fredholm index of hypoelliptic operators on Carnot manifolds.
A graph manifold rational homology 3-sphere W with a left-orderable fundamental group admits a co-oriented taut foliation, though it is unknown whether it admits a smooth co-oriented taut foliation. In this paper we extend the gluing theorem of arXiv:1401.7726 to graph manifold rational homology solid tori and use …
Toroidal 3-manifolds have special group structures that can be shown through specific covers.
problem Characterizing the fundamental groups of toroidal 3-manifolds.
method Proving toroidal 3-manifolds have circularly-orderable fundamental groups by showing they admit finite cyclic covers with left-orderable fundamental groups.
result Toroidal 3-manifolds have circularly-orderable fundamental groups, which can be shown through specific covers.
The paper generalizes Fenchel's theorem for curves with singularities.
problem Proving a generalized Fenchel's theorem for closed curves with singularities.
method Generalization of Fenchel's theorem for closed frontal curves in Euclidean space.
result Total absolute curvature of non-co-orientable closed frontal curves is at least π, with equality conditions.
Summing submanifolds in topological spaces is possible and explicit methods are provided.
problem Summing submanifolds in topological spaces.
method Explicit construction of submanifolds representing the sum of homology classes.
result Explicit construction of submanifolds representing the sum of homology classes.
Contact forms with large systolic ratios found in arbitrary dimensions.
problem Finding contact forms with large systolic ratios in arbitrary dimensions.
method Generalized plug construction and Liouville open books.
result Every co-orientable contact structure admits a contact form with arbitrarily large systolic ratio.
3-manifolds foliar if surfaces minimize Thurston norm, leading to taut foliations.
problem Understanding foliar 3-manifolds via minimizing surfaces.
method Defining double-diamond taut sutured manifolds and analyzing their implications.
result Many slopes are strongly realized in foliations of 3-manifolds, including knot complements.
The paper explores left-orderable groups in branched cyclic covers with taut foliations.
problem Left-orderability of fundamental groups in branched cyclic covers.
method Study of left-orderability in cyclic branched covers of links with co-oriented taut foliations.
result Proves left-orderability for specific types of branched covers and Dehn surgeries.
Study of CR Yamabe problem on toric contact manifolds.
problem CR Yamabe problem on toric contact manifolds.
method Equivalence to boundary value problem for elliptic PDE.
result Many contact toric manifolds admit CR structures with distinct CR Yamabe invariants.
An AH (affine hypersurface) structure is a pair comprising a projective equivalence class of torsion-free connections and a conformal structure satisfying a compatibility condition which is automatic in two dimensions. They generalize Weyl structures, and a pair of AH structures is induced on a co-oriented non-degenera…
The Whitehead link exterior lacks most Euler class taut foliations.
problem Existence of co-orientable taut foliations with specific Euler classes.
method Combining foliation theory techniques and topological obstructions.
result Most lattice points in the dual Thurston ball cannot be Euler classes of co-orientable taut foliations.
Left orderability proven for certain 3-manifolds with specific foliations.
problem Proving left orderability for specific 3-manifolds with taut foliations.
method Analyzing the fundamental group of a 3-manifold with a co-orientable taut foliation.
result The fundamental group of the 3-manifold is left orderable.
The paper proves left-orderability for certain Dehn fillings of pseudo-Anosov mapping tori.
problem Left-orderability of fundamental groups in Dehn fillings of pseudo-Anosov mapping tori.
method Two approaches: one using R-covered foliations and the other using one-sided branching. result All such Dehn fillings have left-orderable fundamental groups.
Taut foliations map leaves to branched 2-sphere covers.
problem Understanding taut foliations in 3-manifolds.
method Map foliation leaves to branched 2-sphere covers.
result Taut foliations are characterized by such maps.
Paper tackles L-space conjecture for knot manifolds, proving equivalence for some properties.
problem Tackles L-space conjecture for knot manifolds, proving equivalence for some properties. method Introduces relative L-space conjecture, characterizes slope detection, uses Heegaard Floer homology, left-orders, and foliations. result Confirms equivalence of CTF and NLS for slope detected knots, identifies exceptional slopes. Introduces a new CR invariant for co-oriented contact structures on closed 3-manifolds
problem Distinguishing contact structures on closed 3-manifolds
method Constructs an invariant μM(ξ) associated with a contact structure ξ and open book decomposition result Shows that the first Chern classes of two tight contact structures on the 3-torus are different
3-manifolds with Heegaard 2 admit taut foliations if their fundamental group is left-orderable.
problem Characterizing 3-manifolds with taut foliations.
method Proving existence of taut foliations based on left-orderability of the fundamental group.
result 3-manifolds with Heegaard genus 2 and left-orderable fundamental group admit co-orientable taut foliations.
We give necessary and sufficient conditions for a closed connected co-orientable contact 3-manifold (M,ξ) to be a standard lens space based on assumptions on the Reeb flow associated to a defining contact form. Our methods also provide rational global surfaces of section for nondegenerate Reeb flows on $(L(p,q),ξ_{…
Persistently foliar knots have a unique foliation property under certain surgeries.
problem Characterizing knots with a specific foliation property under surgeries.
method Analyzing foliations in knot complements and using Dehn surgery.
result Composite knots and nontrivial connected sums of fibered knots are persistently foliar.
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 fa…
Study of knot surgeries and JSJ decompositions to tackle L-space conjecture.
problem Understanding JSJ decompositions and L-space knots through Dehn surgeries. method Slope detection techniques applied to toroidal 3-manifolds and rational surgeries.
result JSJ graphs of certain knot exteriors are rooted intervals, implying left-orderable fundamental groups.
In this paper, we show the existence of (co-oriented) contact structures on certain classes of G2-manifolds, and that these two structures are compatible in certain ways. Moreover, we prove that any seven-manifold with a spin structure (and so any manifold with G2-structure) admits an almost contact structure. We…
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-…
Study on spherical CR manifolds with non-trivial Chern classes.
problem Understanding Chern classes in spherical CR manifolds.
method Construction and proof of constraints on Chern classes.
result Topological obstruction to spherical CR structures on contact manifolds.
New foliations show knot meridians are detectable.
problem Detecting knots in 3-manifolds.
method Constructing co-oriented taut foliations intersecting knot meridians.
result Evidence supports conjecture related to L-space conjecture.
The paper constructs foliations and structures for 3-manifolds with left-orderable fundamental groups.
problem Constructing foliations and structures for 3-manifolds with specific properties.
method A process to produce co-orientable Reebless foliations with transverse (π1(M),R) structures. result The existence of foliations that can extend to taut foliations or R-covered foliations. Detect slopes in toroidal 3-manifolds to prove properties of fundamental groups.
problem Prove properties of fundamental groups of toroidal 3-manifolds.
method Slope detection using left-orders, foliations, and Heegaard Floer homology.
result Toroidal integer homology spheres have left-orderable fundamental groups.
Study on singularity behavior of mean curvature flow with bounded curvature and index.
problem Understanding singularity formation in mean curvature flow with constraints.
method Analyzing flow with bounded mean curvature and Morse index.
result Either mean curvature or Morse index blows up at first singular time.
The paper confirms Ilmanen's conjecture about mean curvature flows.
problem Understanding the behavior of mean curvature flows under type-I conditions.
method Analyzing the convergence of rescaled flows to self-shrinkers with multiplicity one.
result The mean curvature of a closed smooth embedded mean curvature flow in R^3 is of type-I at the first singular time.
Torus decomposition shows foliation detected slopes for glued knot manifolds.
problem Detecting foliation detected slopes in glued knot manifolds.
method Torus decomposition and foliation analysis.
result Gluing knot manifolds identifies rational boundary slopes.
Contact forms on 3-manifolds can have very large systolic ratios.
problem Understanding the systolic ratios of contact forms on 3-manifolds.
method Analyzing the systolic ratio of contact forms defined by co-orientable contact structures.
result Contact forms on any closed 3-manifold can have arbitrarily large systolic ratios.
The motivation of this work is to define cohomology classes in the space of knots that are both easy to find and to evaluate, by reducing the problem to simple linear algebra. We achieve this goal by defining a combinatorial graded cochain complex, such that the elements of an explicit submodule in the cohomology defin…
We show that any co-orientable foliation of dimension two on a closed orientable 3-manifold with continuous tangent plane field can be C0-approximated by both positive and negative contact structures unless all the leaves are simply connected. As applications we deduce that the existence of a taut C0-foliation …
We show that every co--orientable taut foliation F of an orientable, atoroidal 3-manifold admits a transverse essential lamination. If this transverse lamination is a foliation G, the pair F,G are the unstable and stable foliation respectively of an Anosov flow. Otherwise, F admits a pair of transverse very full genuin…
Estimates curvature and radius for constant mean curvature surfaces.
problem Understanding the geometry of surfaces with constant mean curvature.
method Intrinsic curvature and radius estimates for compact disks, applied to complete surfaces.
result Global geometry of constant mean curvature surfaces studied.