arXiv research

Finite volume ends found in quaternionic Kähler manifolds.

Minimal surfaces found in finite volume manifolds.

Entropy rigidity proven for 3D and higher convex projective manifolds.

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

The study counts cusps on finite volume Riemannian manifolds using volume and decay estimates.

We show that complete uniform visibility manifolds of finite volume with sectional curvature $-1 \leq K \leq 0$ have positive simplicial volumes. This implies that their minimal volumes are non-zero.

The paper sets new limits on hyperbolic polyhedra volumes.

3-manifold volumes equal if finite quotients match.

The Cayley hyperbolic space minimizes volume entropy among finite-volume metrics.

A finite-volume hyperbolic 3-manifold geometrically bounds if it is the geodesic boundary of a finite-volume hyperbolic 4-manifold. We construct here an example of non-compact, finite-volume hyperbolic 3-manifold that geometrically bounds. The 3-manifold is the complement of a link with eight components, and its volume…

Investigates simplicial volume over finite fields and compares it with other coefficients.

The study classifies 331 specific 4D polytopes with 7 facets.

We define the ``volume'' contained by pointed $k$-surfaces, first studied by the author in [9], and we show that this volume is always finite. Likewise, we show that the surface area of a pointed $k$-surface is always finite.

Finite types of 4D manifolds with specific curvature, volume, and diameter.

Finite degree formula proven for normalized volumes of klt singularities.

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

Survey on finite-volume hyperbolic four-manifolds.

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

Study on renormalized volume for hyperbolic 3-manifolds, including rank-1 cusps.

Entropy rigidity theorem for negatively curved finite volume manifolds.

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

Study estimates Bergman kernel on hyperbolic surfaces.

The study shows simplicial volume finiteness for certain manifolds with amenable fundamental groups.

We study noncompact, complete, finite volume, negatively curved manifolds $M$. We construct $M$ with infinitely generated fundamental groups in all dimensions $n \geq 2$. We construct $M$ whose cusp cross sections are compact hyperbolic manifolds in all dimension $n\geq 3$. In contrast we show that if sectional curvatu…

Researchers create non-homogeneous finite-volume ends on quaternionic Kähler manifolds.

It is known that the volume function for hyperbolic manifolds of dimension $\geq 3$ is finite-to-one. We show that the number of nonhomeomorphic hyperbolic 4-manifolds with the same volume can be made arbitrarily large. This is done by constructing a sequence of finite-sided finite-volume polyhedra with side-pairings t…

Paper shows 3D hyperbolic manifolds are uniquely identified by their finite groups.

We show that given n>0, there exists a hyperbolic knot K with trivial Alexander polynomial, trivial finite type invariants of order <=n, and such that the volume of the complement of K is larger than n. This contrasts with the known statement that the volume of the complement of a hyperbolic alternating knot is bounded…

Balls are the only volume-constrained critical points of perimeter.

The paper proves bounded domains are biholomorphic to the unit ball.

Study shows simplicial volumes add for certain manifold gluings.

Researchers calculate the volume of Seifert representations for graph manifolds and their covers.

Characterizes metrics with finite total Q-curvature and introduces new volume entropy.

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

The rich theory of Coxeter groups is used to provide an algebraic construction of finite volume hyperbolic n-manifolds. Combinatorial properties of finite images of these groups can be used to compute the volumes of the resulting manifolds. Three examples, in 4,5 and 6-dimensions, are given, each of very small volume, …

Computational evidence supports that SnapPea census manifolds are distinguishable by their finite quotients.

We study a metric version of the simplicial volume on Riemannian manifolds, the Lipschitz simplicial volume, with applications to degree theorems in mind. We establish a proportionality principle and a product inequality from which we derive an extension of Gromov's volume comparison theorem to products of negatively c…

Study shows pinched negatively curved metrics are rarely connected.

Let $M$ be the interior of a connected, oriented, compact manifold $V$ of dimension at least 2. If each path component of $\partial V$ has amenable fundamental group, then we prove that the simplicial volume of $M$ is equal to the relative simplicial volume of $V$ and also to the geometric (Lipschitz) simplicial volume…

A convex projective surface is the quotient of a properly convex open $Ω$ of $\mathbb{P}(\R)$ by a discret subgroup $Γ$ of $\mathrm{SL}_3(\R)$. We give some caracterisations of the fact that a convex projective surface is of finite volume for the Busemann's measure. We deduce of this that if $Ω$ is not a triangle then …

The paper bends hyperbolic manifolds into convex structures.

We compute the space of $L^2$ harmonic forms (outside the middle degrees) on negatively curved Kaehler manifolds of finite volume.

Open manifolds with nonnegative Ricci curvature and Euclidean volume growth have finitely generated fundamental groups.

Study compares volumes of hyperbolic 3-manifolds using Ricci-DeTurck flow.

Paper compares two entropy concepts for finite presentation groups.

Finite volume Coxeter polytopes are quasiperfect and related to finite covolume reflection groups.

Geodesic simplices in pseudo-hyperbolic space get a cohomological treatment.

Finite-volume Ricci solitons with constant-length potential are trivial.