arXiv research

Infinite volumes of Bergman spaces on product manifolds.

Infinite volume found in the thick part of $\mathrm{PSL}_n(\mathbb{R})$-Hitchin-Riemann moduli space.

The paper characterizes sets with infinite hyperbolic convex hull volume.

Infinite hyperbolic manifolds share same perimeter-to-volume ratio.

Extends cohomology theory for infinite volume transformation groups.

In this paper, we prove an equality which involves Reidemeister torsion, complex volume, and Zograf infinite product for closed hyperbolic 3-manifolds.

New insights into ends of quotient spaces and graphs.

Infinite family of hyperbolic 3-manifolds with large volumes.

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

Study intrinsic volume forms on complex hypersurfaces.

In this paper, we prove an equality which involves Reidemeister torsion, complex volume, and Zograf infinite product for hyperbolic 3-manifolds with cusps.

Survey of spectral theory and dynamics for infinite volume hyperbolic manifolds.

The paper bounds eigenvalues of hyperbolic manifolds with infinite volume.

The simplicial volume of non-R^3 contractible 3-manifolds is infinite.

Study of harmonic functions on infinite penny graphs.

New calibration energy measures deviation from calibrated geometry, enabling mean curvature flow in infinite volumes.

Totally geodesic submanifolds in convex cores are properly immersed and have finite volume.

The paper finds infinitely many half-volume CMC hypersurfaces on generic or Ricci-positive manifolds.

We construct two infinite families of ball quotient compactifications birational to bielliptic surfaces. For each family, the volume spectrum of the associated noncompact finite volume ball quotient surfaces is the set of all positive integral multiples of $\frac{8}{3}π^{2}$, i.e., they attain all possible volumes of c…

Negative curvature manifolds have vanishing bounded volume class if and only if Cheeger constant is positive.

We show that the minimal volume entropy of closed manifolds remains unaffected when nonessential manifolds are added in a connected sum. We combine this result with the stable cohomotopy invariant of Bauer-Furuta in order to present an infinite family of four-manifolds with the following properties: 1) They have positi…

The volume of the quantum mechanical state space over $n$-dimensional real, complex and quaternionic Hilbert-spaces with respect to the canonical Euclidean measure is computed, and explicit formulas are presented for the expected value of the determinant in the general setting too. The case when the state space is endo…

Study shows infinite volumes of moduli spaces for certain groups.

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 …

The study proves that certain manifolds can have metrics with specific volume growth.

The paper extends Yamabe flow results to non-compact manifolds with bounded geometry.

Infinite volume requires no atoms at the bottom of the spectrum for certain groups.

We consider the existence of simple closed geodesics or "geodesic knots" in finite volume orientable hyperbolic 3-manifolds. Previous results show that at least one geodesic knot always exists [Bull. London Math. Soc. 31(1) (1999) 81-86], and that certain arithmetic manifolds contain infinitely many geodesic knots [J. …

The work of Jorgensen and Thurston shows that there is a finite number N(v) of orientable hyperbolic 3-manifolds with any given volume v. We show that there is an infinite sequence of closed orientable hyperbolic 3-manifolds, obtained by Dehn filling on the figure eight knot complement, that are uniquely determined by …

We prove the volume conjecture for an infinite family of links called Whitehead chains that generalizes both the Whitehead link and the Borromean rings.

Constructs infinite families of hyperbolic knots satisfying a volume conjecture.

We show that there exist infinitely many commensurability classes of finite volume hyperbolic 3-manifolds whose fundamental group contains a subgroup which is locally free but not free. The main technical tool is the fact that a collection of hyperbolic 3-manifolds of bounded volume contains infinitely many commensurab…

We describe a family of 4-dimensional hyperbolic orbifolds, constructed by deforming an infinite volume orbifold obtained from the ideal, hyperbolic 24-cell by removing two walls. This family provides an infinite number of infinitesimally rigid, infinite covolume, geometrically finite discrete subgroups of the isometry…

In arxiv:1205.1274 Rieck and Yamashita defined the link volume of 3-manifolds and studied some of its basic properties. Many of these properties are similar to the corresponding properties of the hyperbolic volume. In this paper we calculate the link volume of an infinite family of prism manifolds. As a corollary, we s…

The paper proves an infinite double bubble theorem in higher dimensions.

Study circles to understand dynamics and rigidity in homogeneous spaces.

New tools found to create hyperbolic links with lower volume bounds.

This article presents some methods to control the bottom of the spectrum of the Laplacian $λ_0$ on hyperbolic surfaces with infinite volume. Our first result bounds the $λ_0$ of a geometrically finite surface in terms of the geometry of its convex core. We then focus on infinite type periodic hyperbolic surfaces built …

In this paper, we show that Gromov-Thurston's principle works for hyperbolic 3-manifolds of infinite volume and with finitely generated fundamental group. As an application, we have a new proof of Ending Lamination Theorem. Our proof essentially relays only on Maximum Volume Law for hyperbolic 3-simplices.

S. Donaldson introduced a metric on the space of volume forms, with fixed total volume on any compact Riemmanian manifold. With this metric, the space of volume forms formally has non-positive curvature. The geodesic equation is a fully nonlinear degenerate elliptic equation. We solve the geodesic equation and its pert…

Weaving knots are alternating knots with the same projection as torus knots, and were conjectured by X.-S. Lin to be among the maximum volume knots for fixed crossing number. We provide the first asymptotically correct volume bounds for weaving knots, and we prove that the infinite weave is their geometric limit.

We study hyperbolic bongles and find their volumes.

Let $M$ be a convex cocompact acylindrical hyperbolic 3-manifold of infinite volume, and let $M^*$ denote the interior of the convex core of $M$. In this paper we show that any geodesic plane in $M^*$ is either closed or dense. We also show that only countably many planes are closed. These are the first rigidity theore…

Study shows complete affine manifolds have zero simplicial volume.

Quasifuchsian hyperbolic manifolds, or more generally convex co-compact hyperbolic manifolds, have infinite volume, but they have a well-defined ``renormalized'' volume. We outline some relations between this renormalized volume and the volume, or more precisely the ``dual volume'', of the convex core. On one hand, the…

Effective drilling and filling bounds for hyperbolic 3-manifolds.

We introduce a class of regularisable infinite dimensional principal fibre bundles which includes fibre bundles arising in gauge field theories like Yang-Mills and string theory and which generalise finite dimensional Riemannian principal fibre bundles induced by an isometric action. We show that the orbits of regulari…

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