arXiv research

Constructs foliations of critical surfaces for Hawking energy in asymptotically flat initial data sets.

Study local foliations of surfaces with constant mean curvature and constant expansion in space-time.

Absolutely partially hyperbolic surface endomorphisms have a coherent center foliation.

Constructs foliations of lightcones using surfaces of constant spacetime mean curvature.

We study non-compact surfaces obtained by gluing strips $\mathbb{R}\times(-1,1)$ 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.

We deal with minimal surfaces in the unit sphere $S^3$, which are one-parameter families of circles. Minimal surfaces in $\R^3$ foliated by circles were first investigated by Riemann, and a hundred years later Lawson constructed examples of such surfaces in $S^3$. We prove that in $S^3$ there are only two types of mini…

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…

New method shows trapped surfaces form in geodesic foliation.

The aim of this paper is to describe complex foliations on Kahler surfaces.

New foliations found for critical surfaces of Hawking energy, resolving discrepancies.

Minimal hyperbolic foliations on 3-manifolds have non-simply connected generic leaves.

In this paper we study surfaces foliated by a uniparametric family of circles in the homogeneous space Sol$_3$. 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.

Formula for harmonic current dimension on foliated surfaces, extending Brunella's inequality.

Estimates surface count with prescribed foliations.

The paper studies constant harmonic mean curvature surfaces in Schwarzschild spaces, proving they foliate the space.

This paper solves minimal surface equations near Hardt-Simon foliations.

Paper confirms conjecture for PL foliations of codimension 2.

Study dihedral spherical surfaces and their foliations.

A formula calculates the Euler class of foliations using dual graphs.

Introduces zebra structures on surfaces for directional foliation.

The paper proves CMC foliations for quasi-Fuchsian manifolds near 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.

The study characterizes infinite Riemann surfaces and their foliations using quadratic differentials.

The paper constructs foliations of minimal surfaces in negatively curved 3-manifolds.

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.

The study defines Finsler metrics on special surfaces and constructs geodesic currents.

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.

Finite intersection numbers between horizontal foliations of quadratic differentials.

Using the complex parabolic rotations of holomorphic null curves in ${\mathbb{C}}^{4}$, we transform minimal surfaces in Euclidean space ${\mathbb{R}}^{3} \subset {\mathbb{R}}^{4}$ to a family of degenerate minimal surfaces in Euclidean space ${\mathbb{R}}^{4}$. 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.

The paper studies the geometry and topology of a specific foliation on a complex surface.

Study measures volume of foliations on surfaces, finding integrability range.

Paper proves trapped surface formation for Einstein-Maxwell-charged scalar field system.

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.

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…

We describe the space of measured foliations induced on a compact Riemann surface by meromorphic quadratic differentials. We prove that any such foliation is realized by a unique such differential $q$ if we prescribe, in addition, the principal parts of $\sqrt q$ at the poles. This generalizes a theorem of Hubbard and …