Every noncompact surface can be a leaf in a minimal foliation of a 3-manifold.
problem Tackling the minimal foliation structure of noncompact surfaces.
method Using continuous minimal actions of surface groups on the circle to construct minimal foliations.
result Any noncompact oriented surface is homeomorphic to the leaf of a minimal foliation of a closed 3-manifold.
Constructs foliations of critical surfaces for Hawking energy in asymptotically flat initial data sets.
problem Positivity and rigidity of Hawking quasi-local energy in asymptotically flat spacetimes.
method Lyapunov-Schmidt reduction within a Willmore-foliation framework.
result Existence and uniqueness of foliations by Hawking surfaces, positivity and large-sphere limit of Hawking energy.
Study local foliations of surfaces with constant mean curvature and constant expansion in space-time.
problem Characterize surfaces with constant mean curvature and constant expansion in space-time.
method Use Lyapunov Schmidt reduction in an n+1 dimensional manifold to construct and prove the uniqueness of foliations.
result Construct and prove the uniqueness of local foliations of surfaces with constant mean curvature and constant expansion.
Absolutely partially hyperbolic surface endomorphisms have a coherent center foliation.
problem Understanding the dynamics of absolutely partially hyperbolic surface endomorphisms.
method Showed the existence of a center foliation and leaf conjugacy to the linearization.
result Absolutely partially hyperbolic surface endomorphisms have a dynamically coherent center foliation.
Constructs foliations of lightcones using surfaces of constant spacetime mean curvature.
problem Creating foliations of lightcones with specific geometric properties.
method Employing a geometric flow inspired by Huisken-Yau's approach for Riemannian settings.
result Initial data converges exponentially to an STCMC surface under area preserving null mean curvature flow.
We study non-compact surfaces obtained by gluing strips R×(−1,1) with at most countably many boundary intervals along some these intervals. Every such strip possesses a foliation by parallel lines, which gives a foliation on the resulting surface. It is proved that the identity path component of the gro…
Extends Masur's divergence theorem to complex tori and Kummer surfaces.
problem Establishing uniquely ergodic horizontal foliations for geodesic flows on moduli spaces.
method Defined and calculated horizontal foliations and geodesic flows on moduli spaces of Kähler metrics.
result Proved that horizontal foliations are uniquely ergodic if geodesic flows are recurrent.
We deal with minimal surfaces in the unit sphere S3, which are one-parameter families of circles. Minimal surfaces in R3 foliated by circles were first investigated by Riemann, and a hundred years later Lawson constructed examples of such surfaces in S3. We prove that in S3 there are only two types of mini…
New method shows trapped surfaces form in geodesic foliation.
problem Formation of trapped surfaces in Einstein vacuum equation.
method Geodesic foliation, non-integrable PT frame.
result Trapped surfaces form for Einstein vacuum equation.
The aim of this paper is to describe complex foliations on Kahler surfaces.
New foliations found for critical surfaces of Hawking energy, resolving discrepancies.
problem Finding consistent critical surfaces for the Hawking energy in non-totally geodesic spacelike hypersurfaces.
method Constructing a unique local foliation of area constrained critical surfaces of the Hawking energy in the general case of non-totally geodesic spacelike hypersurfaces.
result Discrepancy found in the small sphere limit of the Hawking energy, explained and resolved.
Minimal hyperbolic foliations on 3-manifolds have non-simply connected generic leaves.
problem Characterizing surfaces with minimal hyperbolic foliations.
method Analyzing codimension one foliations on closed 3-manifolds.
result Noncompact surfaces satisfying a specific condition are homeomorphic to the leaf of a minimal foliation with non-simply connected generic leaf.
In this paper we study surfaces foliated by a uniparametric family of circles in the homogeneous space Sol3. We prove that there do not exist such surfaces with zero mean curvature or with zero Gaussian curvature. We extend this study considering surfaces foliated by geodesics, equidistant lines or horocycles in tot…
Study foliations at infinity and constant mean curvature surfaces in quasi-Fuchsian manifolds.
problem Understanding foliations at infinity and constant mean curvature surfaces in quasi-Fuchsian manifolds.
method Using measured foliations and quasi-Fuchsian manifolds, proving the existence and uniqueness of foliations by constant mean curvature surfaces.
result For quasi-Fuchsian manifolds close to the Fuchsian locus, measured foliations at infinity can be uniquely realized and foliated by constant mean curvature surfaces.
Formula for harmonic current dimension on foliated surfaces, extending Brunella's inequality.
problem Calculating the dimension of harmonic currents on foliated complex surfaces.
method Proving a formula involving Furstenberg entropy and Lyapunov exponent.
result Hausdorff dimension of harmonic current is bounded and can be calculated precisely.
The study explores conditions for a manifold to be a leaf of a complete foliation on the open unit ball.
problem Conditions for a manifold to be a leaf of a complete foliation on the open unit ball.
method Analyzes conditions for manifolds to be leaves of complete foliations.
result Provides answers to the question of when a manifold can be a leaf of a complete foliation on the open unit ball.
Estimates surface count with prescribed foliations.
problem Counting square-tiled surfaces with specific foliations.
method Effective estimate with power saving error term.
result Strengthens asymptotic counting formulas.
The paper studies constant harmonic mean curvature surfaces in Schwarzschild spaces, proving they foliate the space.
problem Investigating constant harmonic mean curvature surfaces in Schwarzschild spaces.
method Volume-preserving harmonic mean curvature flow in asymptotically Schwarzschild spaces.
result These surfaces form a foliation of the space outside a large ball.
This paper solves minimal surface equations near Hardt-Simon foliations.
problem Minimal surfaces near Hardt-Simon foliations.
method Uses gluing methods to construct minimal surfaces.
result Constructs minimal surfaces over Hardt-Simon surfaces and near quadratic cones.
Paper confirms conjecture for PL foliations of codimension 2.
problem Confirming Haefliger-Thurston's conjecture for PL foliations.
method Using a version of Mather-Thurston's theorem for PL homeomorphisms.
result Derive new homological properties for PL surface homeomorphisms.
Study dihedral spherical surfaces and their foliations.
problem Characterize dihedral spherical surfaces and their foliations.
method Define and analyze dihedral surfaces and their foliations, introduce geometric decompositions and deformations.
result Determine the dimension of the moduli space for dihedral surfaces.
A formula calculates the Euler class of foliations using dual graphs.
problem Calculating the Euler class of foliations using cooriented branched surfaces.
method Using dual graphs of cooriented branched surfaces to define a simplicial 1-cycle representing the Poincaré dual of the Euler class.
result The formula generalizes previous results and classifies realizable homology classes.
Introduces zebra structures on surfaces for directional foliation.
problem Finding canonical representatives of homotopy classes on surfaces.
method Introduces zebra structures and uses triangulations with edge connections to prove existence.
result Canonical representatives exist for homotopy classes on surfaces with specific triangulations.
The paper proves CMC foliations for quasi-Fuchsian manifolds near the Fuchsian locus.
problem Existence of monotone CMC foliations for quasi-Fuchsian manifolds.
method Analyzes quasi-Fuchsian manifolds near the Fuchsian locus and proves the existence of a unique monotone CMC foliation.
result Proves the existence of a unique monotone CMC foliation for quasi-Fuchsian manifolds in a small neighborhood of the Fuchsian locus.
A global twistor correspondence is established for neutral self-dual conformal structures with alpha-surface foliation when the structure is close to the standard structure on S^2 times S^2. We need to introduce some singularity for the alpha-surface foliation such that the leaves intersect on a fixed two sphere. In th…
A meromorphic quadratic differential on a punctured Riemann surface induces horizontal and vertical measured foliations with pole-singularities. In a neighborhood of a pole such a foliation comprises foliated strips and half-planes, and its leaf-space determines a metric graph. We introduce the notion of an asymptotic …
The study finds surfaces with constant anisotropic mean curvature foliated by circles in Euclidean space.
problem Existence and geometric description of surfaces with constant anisotropic mean curvature.
method Analyzes surfaces with constant anisotropic mean curvature of the Dirichlet energy, proving existence and classifying them.
result Existence and geometric description of surfaces foliated by circles with zero anisotropic mean curvature.
The study characterizes infinite Riemann surfaces and their foliations using quadratic differentials.
problem Characterizing infinite Riemann surfaces and their foliations using quadratic differentials.
method Extending Hubbard-Masur theorem to infinite surfaces and analyzing Jenkins-Strebel differentials.
result Density of Jenkins-Strebel differentials and extension of Kerckhoff's formula for Teichmüller metric.
The paper constructs foliations of minimal surfaces in negatively curved 3-manifolds.
problem Constructing foliations of minimal surfaces in negatively curved 3-manifolds.
method Deformations of totally geodesic foliations, using Grassmann bundle and negatively curved metrics.
result The foliations of minimal surfaces are deformations of totally geodesic foliations.
Piecewise Euclidean structures (identified solid Euclidean polyhedra) on topological 3-dimensional manifolds and pseudo-manifolds are constructed so that they admit pseudo-foliations, a generalized type of foliation. The construction of non-manifold point neighborhoods is done to preserve as much of the geometric, and …
Study shows how certain foliations in unit tangent bundles behave.
problem Characterizing behavior of foliations in unit tangent bundles.
method Analyzing intersections and properties of foliations.
result Certain partially hyperbolic diffeomorphisms are collapsed Anosov flows.
The study defines Finsler metrics on special surfaces and constructs geodesic currents.
problem Defining and studying Finsler metrics on specific geometric structures.
method Defined compatible Finsler distances, studied geodesics, and constructed Liouville currents.
result Constructs a Liouville current for each metric, encoding curve lengths.
We show that for a closed surface of genus at least 5, or a surface of genus at least 2 with at least one marked point, the set of uniquely ergodic foliations and the set of cobounded foliations is path-connected and locally path-connected.
Paper proves uniqueness of specific spacetime surfaces in a lightcone.
problem Proving uniqueness of surfaces of constant spacetime mean curvature in a lightcone.
method Used a fairly generic notion of asymptotic flatness to prove uniqueness.
result Unique foliation by surfaces of constant spacetime mean curvature exists under weaker assumptions.
Finite intersection numbers between horizontal foliations of quadratic differentials.
problem Intersection properties of horizontal foliations in quadratic differentials.
method Joint continuity of intersection number in L1-norm. result Intersection number is finite and jointly continuous.
Using the complex parabolic rotations of holomorphic null curves in C4, we transform minimal surfaces in Euclidean space R3⊂R4 to a family of degenerate minimal surfaces in Euclidean space R4. Applying our deformation to holomorphic null curves in ${…
A meromorphic quadratic differential with poles of order two, on a compact Riemann surface, induces a measured foliation on the surface, with a spiralling structure at any pole that is determined by the complex residue of the differential at the pole. We introduce the space of such measured foliations, and prove that f…
We describe several methods to construct minimal foliations by hyperbolic surfaces on closed 3-manifolds, and discuss the properties of the examples thus obtained.
We explain an error in our paper "A smooth foliation of the 5-sphere by complex surfaces", Ann. Math 156 (2002), p.915-930.
We show the existence of a local foliation of a three dimensional Riemannian manifold by critical points of the Willmore functional subject to a small area constraint around non-degenerate critical points of the scalar curvature. This adapts a method developed by Rugang Ye to construct foliations by surfaces of constan…
Researchers compute de Rham cohomology of geodesic flow foliations on hyperbolic surfaces.
problem Answering a problem posed by Haefliger and Li about geodesic flow foliations.
method Unitary representation theory of PSL(2, R) and Hodge decompositions of de Rham complexes.
result Computed de Rham cohomology of weak stable foliations for various coefficients.
The paper studies the geometry and topology of a specific foliation on a complex surface.
problem Characterizing the geometry and topology of a specific foliation on a complex surface.
method Analyzes the isoperiodic foliation of the stratum ΩM1(1,1,−2), proving each leaf is a surface of infinite genus. result Each leaf is a surface of infinite genus homeomorphic to the Loch Ness monster surface.
Study measures volume of foliations on surfaces, finding integrability range.
problem Volume of combinatorial unit ball of measured foliations on bordered surfaces.
method Analyzes combinatorial moduli spaces and Kontsevich measure.
result Determines range of integrability for (BΣmcomb)s. Paper proves trapped surface formation for Einstein-Maxwell-charged scalar field system.
problem Formation of trapped surfaces in Einstein-Maxwell-charged scalar field system.
method Generalized Christodoulou's approach for spherical symmetry and improved for Minkowskian data.
result Improved bound on trapped surface formation for Minkowskian data.
We extend the unpublished work of M. Handel and R. Miller on the classification, up to isotopy, of endperiodic automorphisms of surfaces. We give the Handel-Miller construction of the geodesic laminations, give an axiomatic theory for pseudo-geodesic lamaniations, show the geodesic laminations satisfy the axioms, and p…
The space of broken hyperbolic structures generalizes the Teichmüller space of a punctured surface, and the space of projectivized broken measured foliations (equivalently, the space of projectivized affine foliations) generalizes the space of projectivized measured foliations. Just as projectivized measured foliations…
Proves a Baum--Bott formula for foliations by curves with logarithmic terms.
problem Analyzing singularities and smoothness in foliations by curves.
method Logarithmic Baum--Bott residues for foliated triples (X,F,D), relating to Poincaré's Problem and GSV indices. result Logarithmic Baum--Bott residues generalize Aleksandrov logarithmic index for vector fields on hypersurfaces.
For any closed oriented surface F of genus at least three, we prove the existence of foliated F-bundles over surfaces such that the signatures of the total spaces are non-zero. We can arrange that the total holonomy of the horizontal foliations preserve a prescribed symplectic form on the fiber. We relate the cohomolog…