arXiv research

Complex hyperbolic triangle groups yield specific 3-manifolds at infinity.

3-manifolds are CR uniformized on spheres, proving a conjecture.

Recently, B.Chow and R.S.Hamilton introduced the cross curvature flow on 3-manifolds. In this paper, we analyze two interesting examples for this new flow. One is on a square torus bundle over a circle, and the other is on a $S^{2}$ bundle over a circle. We show that the global flow exist in both cases. But on the form…

We prove that an open 3-manifold proper homotopy equivalent to a geometrically simply connected polyhedron is simply connected at infinity, generalizing a theorem of V.Poenaru.

Define mass invariant for asymptotically hyperbolic 3-manifolds with toroidal infinity.

Study non-transitive pseudo-Anosov flows using group actions.

We give another proof for a result of Brick stating that the simple connectivity at infinity is a geometric property of finitely presented groups. This allows us to define the rate of vanishing of $\p1i$ for those groups which are simply connected at infinity. Further we show that this rate is linear for cocompact latt…

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

In this paper, we study the limiting behavior of the Brown-York mass and Hawking mass along nearly round surfaces at infinity of an asymptotically flat manifold. Nearly round surfaces can be defined in an intrinsic way. Our results show that the ADM mass of an asymptotically flat 3-manifold can be approximated by some …

Using Morse-Bott techniques adapted to the gauge-theoretic setting, we show that the limiting boundary values of the space of finite energy monopoles on a connected 3-manifold with at least two cylindrical ends provides an immersed Lagrangian submanifold of the vortex moduli space at infinity. By studying the signed in…

If $Γ$ is the fundamental group of a complete finite volume hyperbolic $3$-manifold, Guilloux conjectured that the Borel function on the $\text{PSL}(n,\mathbb{C})$-character variety of $Γ$ should be rigid at infinity, that is it should stay bounded away from its maximum at ideal points. In this paper we prove Guilloux'…

In this article, we are interested in the question whether any complete contractible $3$-manifold of positive scalar curvature is homeomorphic to $\mathbb{R}^{3}$. We study the fundamental group at infinity, $π_{1}^{\infty}$, and its relationship with the existence of complete metrics of positive scalar curvature. We p…

In this paper, we study contact structures on any open 3-manifold V which is the interior of a compact 3-manifold. To do this, we introduce proper contact isotopy invariants called the slope at infinity and the division number at infinity. We first prove several classification theorems for T^2 x [0, \infty), T^2 x R, a…

3-manifolds transform into figure-eight knot complements through complex hyperbolic deformations.

Invariant for 3-manifolds with torus boundary defined.

The work is motivated by a result of Manin, which relates the Arakelov Green function on a compact Riemann surface to configurations of geodesics in a 3-dimensional hyperbolic handlebody with Schottky uniformization, having the Riemann surface as conformal boundary at infinity. A natural question is to what extent the …

Cannon, Swenson, and others have proved numerous theorems about subdivision rules associated to hyperbolic groups with a 2-sphere at infinity. However, few explicit examples are known. We construct an explicit subdivision rule for many 3-manifolds from polyhedral gluings. The manifolds that satisfy the conditions inclu…

We prove that there exists a universal constant $c$ such that any closed hyperbolic 3-manifold admits a triangulation of treewidth at most $c$ times its volume. The converse is not true: we show there exists a sequence of hyperbolic 3-manifolds of bounded treewidth but volume approaching infinity. Along the way, we pro…

The goal of this paper is to investigate the topological structure of open simply-connected 3-manifolds whose scalar curvature has a slow decay at infinity. In particular, we show that the Whitehead manifold does not admit a complete metric, whose scalar curvature decays slowly, and in fact that any contractible comple…

We prove that any real analytic strictly pseudoconvex CR 3-manifold is the boundary (at infinity) of a unique selfdual Einstein metric defined in a neighborhood. The proof uses a new construction of twistor space based on singular rational curves.

Let $(M,g)$ be a complete $(n+1)$-dimensional Riemannian manifold with $2\leq n\leq 6$. Our main theorem generalizes the solution of S.-T. Yau's conjecture on the abundance of minimal surfaces and builds on a result of M. Gromov. Suppose that $(M,g)$ has bounded geometry, or more generally is thick at infinity. Then th…

Study reveals hidden accidental parabolics in complex hyperbolic geometry.

Categorifies Stokes coefficients in Chern-Simons theory models.

Let M be an oriented irreducible 3-manifold with infinite fundamental group and empty or toroidal boundary. Consider any element φin the first cohomology of M with integral coefficients. Then one can define the φ-twisted L^2-torsion function of the universal covering which is a function from the set of positive real nu…

We show that for any C^0 Jordan curve C in the sphere at infinity of H^3, there exists an embedded $H$-plane P_H in H^3 with asymptotic boundary C for any H in (-1,1). As a corollary, we proved that any quasi-Fuchsian hyperbolic 3-manifold M=SxR contains an H-surface S_H in the homotopy class of the core surface S for …

The paper studies nonlinear mass concepts in 3-manifolds with nonnegative scalar curvature.

We define a sutured cobordism category of surfaces with boundary and 3-manifolds with corners. In this category a sutured 3-manifold is regarded as a morphism from the empty surface to itself. In the process we define a new class of geometric objects, called bordered sutured manifolds, that generalize both sutured 3-ma…

In this paper, we establish the general theory of (2+1)-dimensional topological quantum field theory (in short, TQFT) with a Verlinde basis. It is a consequence that we have a Dehn surgery formula for 3-manifold invariants for this kind of TQFT's. We will show that Turaev-Viro-Ocneanu unitary TQFT's obtained from subfa…

Smooth solutions and classification of Dirac-Einstein equations on 3-manifolds.

The study finds a limit on the volume growth of certain 3-manifolds.

We extend the Lyapunov-Schmidt analysis of outlying stable CMC spheres in the work of S. Brendle and the second-named author to the "far-off-center" regime and to include general Schwarzschild asymptotics. We obtain sharp existence and non-existence results for large stable CMC spheres that depend very delicately on th…

We construct the Seiberg-Witten theory on 3-manifolds with Euclidean ends (connected sums of $\R^3$ and a compact manifold) with perturbations which approximate $*dx_3$ at infinity, and describe the structure of the moduli spaces. The setup is inspired by Taubes's program of relating the 4-dimensional Seiberg-Witten in…

Cohomology fractals illustrate complex 3-manifold properties.

We define new higher-order Alexander modules $\mathcal{A}_n(C)$ and higher-order degrees $δ_n(C)$ which are invariants of the algebraic planar curve $C$. These come from analyzing the module structure of the homology of certain solvable covers of the complement of the curve $C$. These invariants are in the spirit of th…

We study R-covered foliations of 3-manifolds from the point of view of their transverse geometry. For an R-covered foliation in an atoroidal 3-manifold M, we show that M-tilde can be partially compactified by a canonical cylinder S^1_univ x R on which pi_1(M) acts by elements of Homeo(S^1) x Homeo(R), where the S^1 fac…

Study on length spectrum of random hyperbolic 3-manifolds.

An unknotting tunnel in a 3-manifold with boundary is a properly embedded arc, the complement of an open neighborhood of which is a handlebody. A geodesic with endpoints on the cusp boundary of a hyperbolic 3-manifold and perpendicular to the cusp boundary is called a vertical geodesic. Given a vertical geodesic in a h…

Study mass and center of mass in flat 3-manifolds, proving existence of foliations.

New tools classify symplectic fillings of contact 3-manifolds.

By using Thurston's bending construction we obtain a sequence of faithful discrete representations ρ_n of the fundamental group of a closed hyperbolic 3-manifold fibering over the circle into the isometry group Iso H^4 of the hyperbolic space H^4. The algebraic limit of ρ_n contains a finitely generated subgroup F whos…

We consider a compact orientable hyperbolic 3-manifold with a compressible boundary. Suppose that we are given a sequence of geometrically finite hyperbolic metrics whose conformal boundary structures at infinity diverge to a projective lamination. We prove that if this limit projective lamination is doubly incompressi…

We consider complete (possibly non-compact) three dimensional Riemannian manifolds (M,g) such that: a) (M,g) is non-collapsed, b) the Ricci curvature of (M,g) is bounded from below, c) the geometry of (M,g) at infinity is not too extreme. Given such initial data (M,g) we show that a Ricci flow exists for a short time i…

We show that the renormalized volume of a quasifuchsian hyperbolic 3-manifold is equal, up to an additive constant, to the volume of its convex core. We also provide a precise upper bound on the renormalized volume in terms of the Weil-Petersson distance between the conformal structures at infinity. As a consequence we…

In the complex-Riemannian framework we show that a conformal manifold containing a compact, simply-connected, null-geodesic is conformally flat. In dimension 3 we use the LeBrun correspondence, that views a conformal 3-manifold as the conformal infinity of a selfdual four-manifolds. We also find a relation between the …

Reconstruct flows and manifolds from their boundary actions on circles.

Minimal annuli constructed in PSL2 via variational method.

The paper embeds CR manifolds into twistor spaces and constructs neutral hyperkähler metrics.

Study curvature loci of 3-manifolds in R^6 and R^5.