arXiv research

The existence of essential closed surfaces surfaces is proven for finite coverings of 3-manifolds that are triangulated by finitely many topological ideal tetrahedra and admit a regular, negatively curved, ideal structure.

We introduced an asymptotic quantity that counts area-minimizing surfaces in negatively curved closed 3-manifolds and show that quantity to only be minimized, among all metrics of sectional curvature less than or equal -1, by the hyperbolic metric.

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

In this note, we develop a condition on a closed curve on a surface or in a 3-manifold that implies that the curve has the property that its length function on the space of all hyperbolic structures on the surface or 3-manifold completely determines the curve. For an orientable surface $S$ of negative Euler characteris…

We show that, for any given 3-manifold M, there are at most finitely many hyperbolic knots K in the 3-sphere and fractions p/q (with q > 22), such that M is obtained by p/q surgery along K. This is a corollary of the following result. If M is obtained by Dehn filling the cusps of a hyperbolic 3-manifold X, where each f…

In this paper we study the rigidity of infinite volume 3-manifolds with sectional curvature $-b^2\le K\le -1$ and finitely generated fundamental group. In-particular, we generalize the Sullivan's quasi-conformal rigidity for finitely generated fundamental group with empty dissipative set to negative variable curvature …

We show that if a closed atoroidal 3-manifold M contains a genuine lamination, then it is group negatively curved in the sense of Gromov. Specifically, we exploit the structure of the non-product complementary regions of the genuine lamination and then apply the first author's Ubiquity Theorem to show that M satisfies …

Study on $k$-surfaces in negatively curved 3-manifolds, focusing on energy and entropy.

We show that hyperbolic 3-manifolds with finitely generated fundamental group are tame, that is the ends are products. We actually work in slightly greater generality with pinched negatively curved manifolds with hyperbolic cusps. This answers a conjecture of Marden and implies the Ahlfors measure conjecture. Applicati…

The classic 2pi-Theorem of Gromov and Thurston constructs a negatively curved metric on certain 3-manifolds obtained by Dehn filling. By Geometrization, any such manifold admits a hyperbolic metric. We outline a program using cross curvature flow to construct a smooth one-parameter family of metrics between the "2pi-me…

We introduce the notion of a topological geodesic in a 3-manifold. Under suitable hypotheses on the fundamental group, for instance word-hyperbolicity, topological geodesics are shown to have the useful properties of, and play the same role in several applications as, geodesics in negatively curved spaces. This permits…

The paper connects currents and entropy in hyperbolic 3-manifolds.

The paper introduces a new flow for Legendrian curves in Sasakian sub-Riemannian 3-manifolds.

New hyperbolic 3-pseudomanifolds with unique properties.

The entropy of minimal surfaces is minimized in hyperbolic manifolds.

Using PL-methods, we prove the Marden's conjecture that a hyperbolic 3-manifold $M$ with finitely generated fundamental group and with no parabolics are topologically tame. Our approach is to form an exhaustion $M_i$ of $M$ and modify the boundary to make them 2-convex. We use the induced path-metric, which makes the s…

Characterizes unknotted curves on Seifert surfaces of twist knots.

The paper solves the Dirichlet problem at infinity for certain negatively curved 3-manifolds.

We study noncompact, complete, finite volume, Riemannian 4-manifolds $M$ with sectional curvature $-1<K<0$. We prove that $π_1 M$ cannot be a 3-manifold group. A classical theorem of Gromov says that $M$ is homeomorphic to the interior of a compact manifold $\M$ with boundary $\partial\barM$. We show that for each $π_1…

Study curves of constant breadth in a specific 3D manifold.

The study examines surfaces in hyperbolic 3-manifolds that become nearly flat.

This paper is devoted to the study of curvature and torsion of almost contact curves in trans-Sasakian 3-Manifolds. The conditions for the frenet curves to be almost contact curves in trans-Sasakian 3-manifolds have been obtained.

New proof shows certain 3D spaces are essentially like infinite space.

Research shows curves in Walker 3-manifolds can lie in flat cylinders.

Totally umbilic surfaces in hyperbolic 3-manifolds are constructed and characterized.

Constructs conformal metrics with negative curvature on manifolds with boundary.

Estimates the dual Thurston norm for foliations on negative curvature 3-manifolds.

We examine three key conjectures in 3-manifold theory: the virtually Haken conjecture, the positive virtual b_1 conjecture and the virtually fibred conjecture. We explore the interaction of these conjectures with the following seemingly unrelated areas: eigenvalues of the Laplacian, and Heegaard splittings. We first gi…

We establish an existence and uniqueness theorem for prime decompositions of theta-curves in $3$-manifolds.

We study the Teichmüller space of negatively curved metrics on a high dimensional manifold, with applications to bundles with negatively curved fibers.

We show that the space of negatively curved metrics of a closed negatively curved Riemannian $n$-manifold, $n\geq 10$, is highly non-connected.

Algorithm transforms weakly negative plumbing trees to negative definite ones.

Survey on embedding 3-manifolds in definite 4-manifolds, focusing on Donaldson's theorem.

We show that the space of nonpositively curved metrics of a negatively curved manifold is highly non connected.

Decides undecidability of equations and first-order theory for Seifert 3-manifold groups.

A simplicial complex is called negatively curved if all its simplices are isometric to simplices in hyperbolic space, and it satisfies Gromov's Link Condition. We prove that, subject to certain conditions, a compact graph of spaces whose vertex spaces are negatively curved 2-complexes, and whose edge spaces are points …

Geometrically interprets symplectic structure in 3-manifold triangulations.

We introduce a geometric evolution equation for 3-manifolds with sectional curvature of one sign which is in some sense dual to the Ricci flow. On a closed 3-manifold with negative sectional curvature, we establish short time existence and a pair of monotonicity formulas for solutions to the flow. One of these formulas…

Chow and Hamilton introduced the cross curvature flow on closed 3-manifolds with negative or positive sectional curvature. In this paper, we study the negative cross curvature flow in the case of locally homogenous metrics on 3-manifolds. In each case, we describe the long time behavior of the solutions of the correspo…

In this paper we study non-negatively curved and rationally elliptic GKM$_4$ manifolds and orbifolds. We show that their rational cohomology rings are isomorphic to the rational cohomology of certain model orbifolds. These models are quotients of isometric actions of finite groups on non-negatively curved torus orbifol…

Simplified proof classifies surfaces using normal curves.

New Einstein metrics found on complex manifolds.

Paper shows regions close to negatively curved metrics are minimal fillings and rigid.

We show that certain negatively twisted torus knots admit Dehn surgeries yielding 3-manifolds with non left-orderable fundamental groups.

We prove that the renormalized volume of almost-Fuchsian hyperbolic $3$-manifolds is non-negative, with equality only for Fuchsian manifolds.

Study on closed curves on negatively curved surfaces, linking number formula, and restrictions.

New Einstein metrics found in curved spaces.

We compute the $L^p$-cohomology spaces of some negatively curved manifolds. We deal with two cases: manifolds with finite volume and sufficiently pinched negative curvature, and conformally compact manifolds.