arXiv research

Study on minimal foliations in 3D manifolds with specific conditions.

In this paper, we investigate the mean curvature flows starting from all non-minimal leaves of the isoparametric foliation given by a certain kind of solvable group action on a symmetric space of non-compact type. We prove that the mean curvature flow starting from each non-minimal leaf of the foliation exists in infin…

Study on harmonic maps on weighted Riemannian foliations.

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…

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

Proof shows cones minimize certain geometric functionals.

The paper solves a 5-manifold foliation problem using a Sasaki-Ricci flow.

Study on Lie groups' conformal foliations and harmonic morphisms.

Minimal conformal foliations on Lie groups are shown to be fibres of harmonic morphisms.

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

Lie foliations with symmetric leaves are smoothly conjugate to homogeneous ones.

New families of Lie groups with special foliations discovered.

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…

In this article, we investigate the stability of leaves of minimal foliations of arbitrary codimension. We also study relations between Jacobi fields and vector fields which preserves a foliation and we use these results to Killing fields.

The purpose of this paper is to show that any extension of a minimal Lie foliation on a compact manifold is a transversaly Riemannian g\h- foliation with trivial normal bundle. This result permits to classify the extensions of a minimal Lie foliation on a compact manifold from the Lie subgroups of its Lie group.

The paper characterizes gaps in minimal foliations on tori using energy criteria.

Paper constructs flows converging to cones and foliations.

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 …

The paper proves the existence of area-minimizing hypersurfaces in AF manifolds of higher dimensions.

Study shows how certain foliations in unit tangent bundles behave.

Minimal hypersurfaces in spheres generated by isoparametric foliations are found.

The study classifies natural almost Hermitian structures on specific Lie groups.

Extends isoparametric foliations and area-minimizing cones in product manifolds.

The paper proves new comparison theorems for sub-Laplacian in foliations with minimal leaves.

The paper proves foliation of area-minimizing hypersurfaces in asymptotically flat manifolds.

The volume of a k-dimensional foliation $\mathcal{F}$ in a Riemannian manifold $M^{n}$ is defined as the mass of image of the Gauss map, which is a map from M to the Grassmann bundle of k-planes in the tangent bundle. Generalizing a construction by Gluck and Ziller, "singular" foliations by 3-spheres are constructed on…

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 ${…

We describe several methods to construct minimal foliations by hyperbolic surfaces on closed 3-manifolds, and discuss the properties of the examples thus obtained.

Proves a principle for one-phase Bernoulli problem minimizers.

Associated with isoparametric foliations of unit spheres, there are two classes of minimal surfaces $-$ minimal isoparametric hypersurfaces and focal submanifolds. By virtue of their rich structures, we find new series of minimizing cones. They are cones over focal submanifolds and cones over suitable products among th…

Let $\mathcal{F}$ be a transversely orientable codimension one minimal foliation without vanishing cycles of a manifold $M$. We show that if the fundamental group of each leaf of $\mathcal{F}$ has polynomial growth of degree $k$ for some non-negative integer $k$, then the foliation $\mathcal{F}$ is without holonomy.

Study conformal foliations on Lie groups, finding new families and harmonic morphisms.

Given a uniform foliation by Gromov hyperbolic leaves on a $3$-manifold, we show that the action of the fundamental group on the universal circle is minimal and transitive on pairs of different points. We also prove two other results: we prove that general uniform Reebless foliations are $\mathbb{R}$-covered and we giv…

It is well-known that any isotopically connected diffeomorphism group $G$ of a manifold determines uniquely a singular foliation $\F_G$. A one-to-one correspondence between the class of singular foliations and a subclass of diffeomorphism groups is established. As an illustration of this correspondence it is shown that…

New Lie groupoid and algebroid constructed for octonionic Hopf foliation.

In this paper, we prove Kirchberg inequalities for any kahler spin foliations. Their limiting cases are then characterized as being transversal minimal Einstein foliations. The key point is to introduce the transversal kahlerian twistor operators.

We obtain geometric characterizations of isospectral minimal Riemannian Legendre foliations on compact Sasakian manifolds of constant $φ$-sectional curvature.

This paper deals with the question of analytic continuation of holonomy germs of holomorphic foliations. We prove that for a quasi-minimal Riccati foliation of the complex projective plane, any holonomy germ of the foliation between complex projective lines can be analytically continued along a generic Brownian path.

In this paper, we prove that minimal hypersurfaces when $n\geq 3$ and nonzero constant mean curvature hypersurfaces when $n\geq2$ foliated by spheres in parallel horizontal hyperplanes in ${\mathbb{H}}^n \times \mathbb{R}$ must be rotationally symmetric.

Study mean curvature flow in hyperbolic 3-manifolds, proving foliations and existence of minimal surfaces.

On a Riemannian 2-torus $(T^2,g)$ we study the geodesic flow in the case of low complexity described by zero topological entropy. We show that this assumption implies a nearly integrable behavior. In our previous paper \cite{GK} we already obtained that the asymptotic direction and therefore also the rotation number ex…

Study compatible and associated metrics for contact-symplectic structures, showing geodesic integral curves and minimal leaf properties.

In this paper, we prove that the $3$-sphere endowed with an arbitrary Riemannian metric either contains at least two embedded minimal $2$-spheres or admits an optimal foliation by $2$-spheres. This generalizes recent results by Haslhofer-Ketover (Duke Math. J. 2019), where the existence of optimal foliations and minima…

In this article, we show that, for any compact 3-manifold, there is a $C^{1}$ volume-minimizing one-dimensional foliation. More generally, we show the existence of mass-minimizing rectifiable sections of sphere bundles without isolated "pole points" in the base manifold. This same analysis is used to show that the exam…

The paper classifies natural almost Hermitian structures on Lie groups with minimal conformal leaves.

Minimal hypersurfaces scarring along a fixed one in certain manifolds.

Study foliations on Riemannian manifolds with specific vector fields, focusing on geometric properties.