arXiv research

Finite volume ends found in quaternionic Kähler manifolds.

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

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

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.

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

We prove that every complete non-compact manifold of finite volume contains a (possibly non-compact) minimal hypersurface of finite volume. The main tool is the following result of independent interest: if a region $U$ can be swept out by a family of hypersurfaces of volume at most $V$, then it can be swept out by a fa…

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…

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.

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

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

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

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…

In this paper we prove: if a bounded domain with $C^2$ boundary covers a manifold which has finite volume with respect to either the Bergman volume, the Kähler-Einstein volume, or the Kobayashi-Eisenman volume, then the domain is biholomorphic to the unit ball. This answers an old question of Yau. Further, when the dom…

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 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, …

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…

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

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…

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

We define and study the renormalized volume for geometrically finite hyperbolic $3$-manifolds, including with rank-$1$ cusps. We prove a variation formula, and show that for certain families of convex co-compact hyperbolic metrics $g_\eps$ degenerating to a geometrically finite hyperbolic metric $g_0$ with rank-$1$ cus…

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.

The paper contains a new proof that a complete, non-compact hyperbolic $3$-manifold $M$ with finite volume contains an immersed, closed, quasi-Fuchsian surface.

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

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

We give an upper bound for the growth of homology torsions of finite coverings of irreducible 3-manifolds with tori boundary in terms of hyperbolic volume.

In this article we study the spectrum of totally geodesic surfaces of a finite volume hyperbolic 3-manifold. We show that for arithmetic hyperbolic 3-manifolds that contain a totally geodesic surface, this spectrum determines the commensurability class. In addition, we show that any finite volume hyperbolic 3-manifold …

Estimates for VPMCF show ancient MCF solutions and finite-time behavior.

We establish the proportionality principle between the Riemannian volume and locally finite simplicial volume for Q-rank 1 locally symmetric spaces covered by products of hyperbolic spaces, giving the first examples for manifolds whose cusp groups are not necessarily amenable. Also, we give a simple direct proof of the…

We consider properly immersed finite topology minimal surfaces S in complete finite volume hyperbolic 3-manifolds N, and in M x S(1), where M is a complete hyperbolic surface of finite area. We prove S has finite total curvature equal to 2πtimes the Euler characteristic of S, and we describe the geometry of the ends of…

The study establishes equivalence of conditions on metric manifolds with finite volume.

Uniform linear bounds on volume changes in 3D hyperbolic spaces.

Torus covers have controlled volume and diameter under curvature and diameter bounds.

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

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…

We compare the volume of a hyperbolic 3-manifold $M$ of finite volume and the complexity of its fundamental group.

Study on minimal submanifolds with finite curvature in Euclidean space.