arXiv research

14 homogeneous convex foliations of degree 5 found on complex projective plane.

5 homogeneous convex foliations of degree 4 found on complex projective plane.

A new proof simplifies the classification of convex foliations of degree 2 on complex projective plane.

Study foliations in PSL(4,R)-Teichmüller theory, proving two invariant foliations.

Compact foliations preserve entropy if leaves are strictly convex projective.

Generalizes Giroux's result to higher dimensions and applies to contact manifolds.

Foliation of star-shaped polygons with fixed perimeter and area.

The paper introduces foliated open books for contact 3-manifolds with boundary foliations.

New foliations at infinity for quasi-Fuchsian manifolds near the Fuchsian locus are uniquely determined.

This article describes the following results which relate to each other; i) convergence of high dimensional contact structure to codimension one foliation with Reeb component, ii) relation between Nil-type and Sol-type contact submanifolds of S^5, iii) definition of convex Thurston-Bennequin inequality, and iv) general…

A singular riemannian foliation on a complete riemannian manifold is said to be riemannian if each geodesic that is perpendicular at one point to a leaf remains perpendicular to every leaf it meets. The singular foliation is said to admit sections if each regular point is contained in a totally geodesic complete immers…

Open, connected, saturated sets W without holonomy in codimension one foliations play key roles as fundamental building blocks. Here, for the case of foliated 3-manifolds, we produce a finite system of closed, convex, non-overlapping polyhedral cones in the first cohomology of W with real coefficients such that the iso…

Study on projective structures and their foliations on surfaces.

Study on mapping class groups of non-orientable surfaces, proving some conjectures and refuting others.

The paper studies affine manifolds with linear foliations and their topological properties.

This note provides an alternative proof of a result of Labourie. We show that the two complements of the convex core of a three dimensional quasi-fuchsian hyperbolic manifold may be foliated by embedded hypersurfaces of constant Gaussian curvature.

In this article we give a geometric interpretation of the Hitchin component for PSL(4,R) in the representation variety of a closed oriented surface of higher genus. We show that representations in the Hitchin component are precisely the holonomy representations of properly convex foliated projective structures on the u…

The study shows how to foliate convex hypersurfaces in affine space with constant curvature.

Unique CMC foliation in Minkowski space solved.

Study foliations at infinity and constant mean curvature surfaces in quasi-Fuchsian manifolds.

This submission has been withdrawn by the author and superseded by arXiv:0804.0744.

Non-convex extremal length found in surface metrics.

We prove the local invertibility, up to potential fields, and stability of the geodesic X-ray transform on tensor fields of order 1 and 2 near a strictly convex boundary point, on manifolds with boundary of dimension n>=3. We also present an inversion formula. Under the condition that the manifold can be foliated with …

We characterize convex isoperimetric sets in the Heisenberg group endowed with horizontal perimeter. We first prove Sobolev regularity for a certain class of vector fields in the plane with bounded variation, related to the curvature equations. Then, by an approximation-reparameterization argument, we show that the bou…

The study introduces surfaces in quasi-Fuchsian manifolds and their asymptotic properties.

We develop some pluripotential theoretic techniques for the transversally holomorphic foliation of a Sasakian manifold. We prove the convexity of the K-energy along weak geodesics for Sasakian manifolds. This implies that the K-energy is bounded below if a constant scalar curvature structure exists with those metrics m…

The paper generalizes Birkhoff's conjecture for billiards.

This paper studies global webs on the projective plane with vanishing curvature. The study is based on an interplay of local and global arguments. The main local ingredient is a criterium for the regularity of the curvature at the neighborhood of a generic point of the discriminant. The main global ingredient, the Lege…

We prove that for any convex globally hyperbolic maximal (GHM) anti-de Sitter (AdS) 3-dimensional space-time $N$ with particles (cone singularities of angles less than $π$ along time-like curves), the complement of the convex core in $N$ admits a unique foliation by constant Gauss curvature surfaces. This extends, and …

In 3D affine space, unique foliation of domains by surfaces with constant Gaussian curvature.

Outer billiards studied in complex hyperbolic plane, proving smooth and symplectic properties.

We prove that any hyperbolic end with particles (cone singularities along infinite curves of angles less than $π$) admits a unique foliation by constant Gauss curvature surfaces. Using a form of duality between hyperbolic ends with particles and convex globally hyperbolic maximal (GHM) de Sitter spacetime with particle…

Innovative contact invariant derived from Heegaard Floer homology.

In recent work, the notion of Double Convexity for a foliation of a conical null hypersurface was introduced to give a proof, if satisfied, of the Null Penrose Inequality. Double Convexity constrains the geometry of a Marginally Outer Trapped Surface (MOTS), called a quasi-round MOTS. In the first part of this paper, f…

We study the existence of surfaces with constant or prescribed Gauss curvature in certain Lorentzian spacetimes. We prove in particular that every (non-elementary) 3-dimensional maximal globally hyperbolic spatially compact spacetime with constant non-negative curvature is foliated by compact spacelike surfaces with co…

The paper explores symplectic foliations and their leaves on manifolds.

We prove an asymptotic formula for the number of integer points in a family of bounded domains in the Euclidean space with smooth boundary, which remain unchanged along some linear subspace and stretch out in the directions, orthogonal to this subspace. A more precise estimate for the remainder is obtained in the case …

The study finds at least 2 free-boundary minimal disks in convex 3-balls.

We define an explicit quasi-local mass functional which is non-decreasing along all foliations (satisfying a convexity assumption) of null cones. We use this new functional to prove the null Penrose conjecture under fairly generic conditions.

Proves Birkhoff-Poritsky conjecture for centrally-symmetric billiards.

The basic cohomology of a Riemannian foliation on a complete manifold with all leaves closed is the cohomology of the leaf space. In this paper we introduce various methods to compute the basic cohomology in the presence of both closed and non-closed leaves in the simply-connected case (or more generally for Killing fo…

New method uses broken scattering to uniquely identify Finsler manifolds.

Under a convexity assumption on the boundary we solve a local inverse problem, namely we show that the geodesic X-ray transform can be inverted locally in a stable manner; one even has a reconstruction formula. We also show that under an assumption on the existence of a global foliation by strictly convex hypersurfaces…

Maps on infinite-type surfaces linked to 3-manifold flows.

Let $D$ be a bounded logarithmically convex complete Reinhardt domain in $\mathbb{C}^n$ centered at the origin. Generalizing a result for the one-dimensional case of the unit disk, we prove that the $C^*$-algebra generated by Toeplitz operators with bounded measurable separately radial symbols (i.e., symbols depending …

Study ancient solutions to free boundary mean curvature flow in convex manifolds.

Geometrically, Kostant's Convexity Theorem is extended to submetries with a fat section.

Study describes periodic controls in step 2 sub-Finsler problems on Carnot groups.