arXiv research

The paper defines and analyzes a volume invariant for 3-manifolds.

Minimal volume entropy vanishes or is positive under certain fiber growth conditions.

The study examines conditions for minimal volume entropy of simplicial complexes.

CAT(0) spaces with small volume growth are homeomorphic to Euclidean space.

The goal of this article is to establish estimates involving the Yamabe minimal volume, mixed minimal volume and some topological invariants on compact 4-manifolds. In addition, we provide topological sphere theorems for compact submanifolds of spheres and Euclidean spaces, provided that the full norm of the second fun…

Hyperbolic volume correlates with chemical properties of fullerenes.

Constructs manifolds with infinite Betti numbers and close to quadratic volume growth.

We construct pairs of non-isometric hyperbolic 3-orbifolds with the same topological type and volume. Topologically these orbifolds are mapping tori of pseudo-Anosov maps of the surface of genus 2, with singular locus a fibred (hyperbolic) link with five components.

Improved lower bounds on volumes of hyperbolic 3-manifolds with specific topologies.

The paper proves finite topological type theorems for open manifolds with non-negative Ricci curvature and almost maximal local rewinding volume.

The paper explores generalizations of Mirzakhani's recursion and computes volumes for physical gravity models.

In this paper, we prove that the systolic volume of a closed aspherical 3-manifold is bounded below in terms of complexity. Systolic volume is defined as the optimal constant in a systolic inequality. Babenko showed that the systolic volume is a homotopy invariant. Moreover, Gromov proved that the systolic volume depen…

In this paper, we study the topology of complete noncompact Riemannian manifolds with asymptotically nonnegative Ricci curvature and large volume growth. We prove that they have finite topological types under some curvature decay and volume growth conditions. We also generize it to the manifolds with $k$-th asymptotica…

Compactness theorem for Riemannian manifolds with volume and curvature bounds.

Study reveals failure of uniqueness in dynamical invariants for 3D volume-preserving diffeomorphisms.

Study volumes of Klein surfaces, extending Mirzakhani's recursion.

No CMC surfaces exist in certain hyperbolic 3-manifolds.

Estimates open sets for fibrations, leading to volume vanishing results.

Classifies gravitational instantons with quadratic volume growth.

We study the volume growth of hyperkaehler manifolds of type $A_{\infty}$ constructed by Anderson-Kronheimer-LeBrun and Goto. These are noncompact complete 4-dimensional hyperkaehler manifolds of infinite topological type. These manifolds have the same topology but the hyperkaehler metrics are depends on the choice of …

Withdrawn and replaced by two related manuscripts: (1) "Stabilization in the braid groups I:MTWS", published in Geometry and Topology Volume 10 (2006), 413-540, arXiv:math.GT/0310279, and (2) "Stabilization in the braid groups II: Transversal simplicity of knots", Geometry and Topology Volume 10 (2006), to appear, arXi…

We prove there exists a compact embedded minimal surface in a complete finite volume hyperbolic $3$-manifold $\mathcal{N}$. We also obtain a least area, incompressible, properly embedded, finite topology, $2$-sided surface. We prove a properly embedded minimal surface of bounded curvature has finite topology. This dete…

Novel M-theory approach classifies topological phases of matter.

Study minimizers of quasi-perimeters in RCD spaces with volume constraints.

This is a survey of topological properties of open, complete nonpositively curved manifolds which may have infinite volume. Topics include topology of ends, restrictions on the fundamental group, as well as a review of known examples.

We study the geometry and topology of Riemannian 3-orbifolds which are locally volume collapsed with respect to a curvature scale. We show that a sufficiently collapsed closed 3-orbifold without bad 2-suborbifolds either admits a metric of nonnegative sectional curvature or satisfies Thurston's Geometrization Conjectur…

In this paper we determine the topology of three-dimensional complete orientable Riemannian manifolds with a uniform lower bound of sectional curvature whose volume is sufficiently small.

The volume conjecture states that for a hyperbolic knot K in the three-sphere S^3 the asymptotic growth of the colored Jones polynomial of K is governed by the hyperbolic volume of the knot complement S^3\K. The conjecture relates two topological invariants, one combinatorial and one geometric, in a very nonobvious, no…

New topological restrictions found for spaces with nonnegative Ricci curvature.

Study of $p$-adic simplicial volumes and their properties.

This research extends topological recursion to hyperbolic surfaces with tight boundaries and conical defects.

We prove that the Whitehead link complement and the (-2, 3, 8) pretzel link complement are the minimal volume orientable hyperbolic 3-manifolds with two cusps, with volume 3.66... = 4 x Catalan's constant. We use topological arguments to establish the existence of an essential surface which provides a lower bound on vo…

Finite simplicial complexes dominate certain manifolds with a bounded number of simplices.

We study the Masur-Veech volumes $MV_{g,n}$ of the principal stratum of the moduli space of quadratic differentials of unit area on curves of genus $g$ with $n$ punctures. We show that the volumes $MV_{g,n}$ are the constant terms of a family of polynomials in $n$ variables governed by the topological recursion/Virasor…

This paper contains a purely topological theorem and a geometric application. The topological theorem states that if M is a simple closed orientable 3-manifold such that π_1(M) contains a genus g surface group and H_1(M;Z/2Z) has rank at least 4g-1 then M contains a closed incompressible surface of genus at most g. Thi…

Study shows complete affine manifolds have zero simplicial volume.

Let (M,g) be a compact Riemannian manifold of hyperbolic type, i.e M is a manifold admitting another metric of strictly negative curvature. In this paper we study the geodesic flow restricted to the set of geodesics which are minimal on the universal covering. In particular for surfaces we show that the topological ent…

Study on hyperbolic surfaces' volumes, proving asymptotic expansion for high genus.

Study geometric and topological properties of Finsler manifolds with weighted Ricci curvature bounds.

We show that there are topological obstructions for a noncompact manifold to admit a Riemannian metric with quadratic curvature decay and a volume growth which is slower than that of Euclidean space of the same dimension.

We show that every volume preserving codimension one Anosov flow on a closed Riemannian manifold of dimension greater than three admits a global cross section and is therefore topologically conjugate to a suspension of a linear toral automorphism. This proves a conjecture of Verjovsky from the 1970's in the volume pres…

In this paper we study the topology of conformally compact Einstein 4-manifolds. When the conformal infinity has positive Yamabe invariant and the renormalized volume is also positive we show that the conformally compact Einstein 4-manifold will have at most finite fundamental group. Under the further assumption that t…

For a closed, strictly convex projective manifold of dimension $n\geq 3$ that admits a hyperbolic structure, we show that the ratio of Hilbert volume to hyperbolic volume is bounded below by a constant that depends only on dimension. We also show that for such spaces, if topological entropy of the geodesic flow goes to…

We give an efficient simplicial formula for the volume and Chern-Simons invariant of a boundary-parabolic PSL(2,C)-representation of a tame 3-manifold. If the representation is the geometric representation of a hyperbolic 3-manifold, our formula computes the volume and Chern-Simons invariant directly from an ideal tria…

This paper investigates the relationship between the topology of hyperbolizable 3-manifolds M with incompressible boundary and the volume of hyperbolic convex cores homotopy equivalent to M. Specifically, it proves a conjecture of Bonahon stating that the volume of a convex core is at least half the simplicial volume o…

For a knot K in S^3 we construct according to Casson--or more precisely taking into account Lin and Heusener's further works--a volume form on the SU(2)-representation space of the group of K. We prove that this volume form is a topological knot invariant and explore some of its properties.

The simplicial volume introduced by Gromov provides a topologically accessible lower bound for the minimal volume. Lafont and Schmidt proved that the simplicial volume of closed, locally symmetric spaces of non-compact type is positive. In this paper, we present a generalization of this result to certain non-compact lo…

Suppose $\bar{M}$ is a compact connected odd-dimensional manifold with boundary, whose interior $M$ comes with a complete hyperbolic metric of finite volume. We will show that the $L^2$-topological torsion of $\bar{M}$ and the $L^2$-analytic torsion of the Riemannian manifold $M$ are equal. In particular, the $L^2$-top…