New insights into ends of quotient spaces and graphs.
problem Understanding the ends of quotient spaces and graphs.
method Analyzing infinite volume ends of quotient spaces and graphs.
result Quotient spaces and graphs have exactly one infinite volume end under certain conditions.
Any action of a group Γ on H3 by isometries yields a class in degree three bounded cohomology by pulling back the volume cocycle to Γ. We prove that the bounded cohomology of finitely generated Kleinian groups without parabolic elements distinguishes the asymptotic geometry of geometrically infinite ends…
Researchers create non-homogeneous finite-volume ends on quaternionic Kähler manifolds.
problem Constructing non-homogeneous quaternionic Kähler manifolds with finite volume ends.
method Using cohomogeneity one deformation of symmetric spaces, the researchers constructed manifolds with specific fundamental groups.
result The constructed manifolds are aspherical and have finite volume ends, not locally homogeneous.
The paper explores curvature-free effects in manifolds with volume growth and ends-counting.
problem Investigating curvature-free effects in manifolds with volume growth and ends-counting.
method Establishing two main theorems about volume growth and ends-counting.
result Proves the existence of smooth bounded mean-concave exhaustion and escaping geodesic lines.
Let Mm, with m≥3, be an m-dimensional complete noncompact manifold isometrically immersed in a Hadamard manifold Mˉ. Assume that the mean curvature vector has finite Lp-norm, for some 2≤p≤m. We prove that each end of M must either have finite volume or be non-parabolic.
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.
Every connected, weighted graph with non-negative curvature has exactly two ends.
problem Characterizing the structure of connected, weighted graphs with non-negative curvature.
method Extremal Lipschitz extensions, variational principle, study of harmonic functions.
result Every salami has exactly two ends and no vertices with positive curvature.
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.
Paper explains dynamics of homeomorphisms to mapping tori geometry.
problem Understanding dynamics of end-periodic homeomorphisms.
method Illustration-driven overview of recent results.
result Analogue of Brock's theorem for infinite-type surfaces.
We show that a noncompact manifold with bounded sectional curvature, whose ends are sufficiently Gromov-Hausdorff close to rays, has a finite dimensional space of square-integrable harmonic forms. In the special case of a finite-volume manifold with pinched negative sectional curvature, we show that the essential spect…
New 4-manifolds found without certain Einstein metrics, using Seiberg-Witten theory.
problem Constructing 4-manifolds without specific Einstein metrics.
method Using Seiberg-Witten theory and constructing solutions on noncompact manifolds.
result Infinitely many examples of 4-manifolds without cusped asymptotically hyperbolic Einstein metrics.
Finite volume ends found in quaternionic Kähler manifolds.
problem Existence of quaternionic Kähler manifolds with finite volume ends.
method Proof in all dimensions 4m≥4. result Existence of quaternionic Kähler manifolds with finite volume ends.
Infinite volumes of Bergman spaces on product manifolds.
problem Volume calculation of Bergman spaces on product manifolds.
method Calabi and Mabuchi volumes, inspired by Shiffman-Zelditch conjecture.
result Infinite volumes of Bergman spaces on product manifolds.
Constructs knots from 3-manifolds with specified geometric limits.
problem Build knots from 3-manifolds with specific geometric properties.
method Constructs explicit families of knots converging to specified 3-manifolds.
result Constructs knots from geometrically finite 3-manifolds with one end.
We study totally geodesic planes in hyperbolic 3-manifolds M having incompressible core and degenerate ends. We prove a Ratner-type phenomenon: a closed minimal PSL(2,R)−invariant subset of M is either an immersed totally geodesic surface or all of M. We also show that for an arbitrary infinite volume hyperboli…
Veech groups uniformize Teichmüller geodesic curves in Riemann moduli space. Recently, examples of infinitely generated Veech groups have been given. We show that these can even have infinitely many cusps and infinitely many infinite ends. We further show that examples exist for which each direction of an infinite end …
Infinite volume found in the thick part of PSLn(R)-Hitchin-Riemann moduli space.
problem Proving infinite volume in the thick part of PSLn(R)-Hitchin-Riemann moduli space. method Employing Goldman flows and internal sequences to find an infinite series of subsets of identical volume.
result Infinite total Atiyah--Bott--Goldman volume for n>2. The paper characterizes sets with infinite hyperbolic convex hull volume.
problem Characterizing sets with infinite hyperbolic convex hull volume.
method Geometric conditions and self-similar sets.
result Characterizes continua and planar self-similar sets with infinite hyperbolic convex hull volume.
Two results on end spaces of infinite type surfaces, answering questions about their topology and equivalence.
problem Topology and equivalence of end spaces of infinite type surfaces.
method Examples and Tsankov's argument to show equivalence relation.
result Examples of infinite type surfaces with end spaces that are not self-similar but have a unique maximal type.
We prove the existence of nonperiodic, properly embedded minimal surfaces in R2×S1 with genus zero, infinitely many ends and one limit end (in particular, they have infinite total curvature).
Infinite-genus surfaces have many isospectral hyperbolic structures.
problem Finding many isospectral hyperbolic structures on infinite-genus surfaces.
method Constructing families of isospectral hyperbolic structures on infinite-type surfaces without planar ends.
result Uncountable families of isospectral and quasiconformally distinct hyperbolic structures on infinite-genus surfaces with self-similar end spaces.
Lower bound on volumes of special mapping tori.
problem Calculating the minimum volume of compactified mapping tori.
method Using strongly irreducible end-periodic homeomorphisms and properties of pants graphs.
result Volume of compactified mapping tori is comparable to the translation length of the homeomorphism on pants graphs.
Study shows certain infinite-ended groups are not quasi-isometrically rigid.
problem Understanding when infinite-ended groups are quasi-isometrically rigid.
method Combining results on subgroups and hyperbolic groups, adapting Whyte's argument.
result Proves certain infinite-ended groups are not quasi-isometrically rigid.
Infinite hyperbolic manifolds share same perimeter-to-volume ratio.
problem Finding hyperbolic manifolds with a fixed perimeter-to-volume ratio.
method Constructing infinitely many hyperbolic manifolds with nonempty boundaries.
result Existence of incommensurable hyperbolic manifolds with a fixed perimeter-to-volume ratio.
Let M4n be a complete quaternionic Kähler manifold with scalar curvature bounded below by −16n(n+2). We get a sharp estimate for the first eigenvalue λ1(M) of the Laplacian which is λ1(M)≤(2n+1)2. If the equality holds, then either M has only one end, or M is diffeomorphic to R×N w…
Extends cohomology theory for infinite volume transformation groups.
problem Generalizing cohomology for infinite volume transformation groups.
method Introduces norm-controlled cohomology as a generalization of bounded cohomology.
result Establishes norm-controlled cohomology 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.
Accessible groups with infinitely many ends have infinitely many twisted conjugacy classes.
problem Characterizing groups with infinitely many ends and their conjugacy classes.
method Analyzing accessible groups and relatively hyperbolic groups to deduce properties.
result Groups with infinitely many ends have infinitely many twisted conjugacy classes.
Infinite family of hyperbolic 3-manifolds with large volumes.
problem Finding large volume 3-manifolds.
method 2-fold branched covers of 3-manifolds M.
result Existence of hyperbolic 3-manifolds with arbitrarily large volume.
Grimaldi-Pansu metrics are constructed for manifolds with multiple ends.
problem Volume growth on manifolds with more than one end.
method Constructing Riemannian metrics with bounded geometry and uniform bounds for volume growth.
result Uniform bounds for volume growth of Grimaldi-Pansu metrics in certain manifolds.
Constructs manifolds with infinite Betti numbers and close to quadratic volume growth.
problem Understanding if manifolds with specific curvature bounds and volume growth must be of finite topological type.
method Constructs a family of (2+n)−dimensional open manifolds with positive Ricci curvature and sectional curvature bounds. result Volume growth can be arbitrarily close to quadratic, and Betti numbers are infinite.
Defines complex for infinite-type surfaces with non-planar ends.
problem No finite analog for surfaces with non-planar ends.
method Defined a complex Γk(S) for infinite-type surfaces. result Automorphism group of Γk(S) is isomorphic to the extended mapping class group. Study intrinsic volume forms on complex hypersurfaces.
problem Computing volume functionals on pseudoconvex hypersurfaces.
method Compute first and second variation formulae, explore infinite dimensional aspects.
result Discuss possible analogues of the affine isoperimetric inequality.
Upper bound on 3-manifold volumes from surface homeomorphisms.
problem Bounding volumes of 3-manifolds from surface homeomorphisms.
method Using end-periodic homeomorphisms and pants graphs.
result Upper bound on infimal hyperbolic volume is asymptotically sharp.
In this paper, we prove an equality which involves Reidemeister torsion, complex volume, and Zograf infinite product for hyperbolic 3-manifolds with cusps.
For any m > 0, we construct properly embedded minimal surfaces in H^2 x R with genus zero, infinitely many vertical planar ends and m limit ends. We also provide examples with an infinite countable number of limit ends. All these examples are vertical bi-graphs.
Survey of spectral theory and dynamics for infinite volume hyperbolic manifolds.
problem Understanding infinite volume asymptotically hyperbolic manifolds.
method Survey of geometry, spectral theory, dynamics, and quantum/classical mechanics.
result Recent results, ideas, and conjectures discussed.
The study examines heat kernel bounds on Riemannian manifolds with an end.
problem Estimating heat kernel on Riemannian manifolds with an end.
method Investigates heat kernel estimates of the form pt(x,x)≥cxt−α for large enough t. result Establishes bounds on the form pt(x,x)≥cxt−α for large enough t. Study of ends of complete gradient Schouten solitons, showing finitely many ends for shrinking and connected infinity for expanding ones.
problem Characterizing the ends of complete gradient Schouten solitons.
method Analysis of ends without additional assumptions, focusing on shrinking and expanding cases.
result Finitely many ends for shrinking Schouten solitons and connected infinity for expanding ones.
The paper bounds eigenvalues of hyperbolic manifolds with infinite volume.
problem Bounding eigenvalues of geometrically finite hyperbolic manifolds of infinite volume.
method Provided a lower bound on the kth eigenvalue of the Laplace-Beltrami operator by the kth eigenvalue of a neighborhood of the thick part of the convex core.
result Recovered a theorem bounding the bottom eigenvalue from below by a specific formula involving the volume of the 1-neighborhood of the convex core.
Study infinite genus surfaces and Schottky groups for uniformization.
problem Investigate infinite genus surfaces and Schottky groups for uniformization.
method Definitions and proofs for infinite genus surfaces and Schottky groups, showing uniformization by Schottky groups.
result Infinite genus surfaces and handlebodies can be topologically and quasiconformally uniformized by Schottky groups.
The study shows simplicial volume finiteness for certain manifolds with amenable fundamental groups.
problem Determining the simplicial volume of manifolds with specific properties.
method Analyzing the fundamental group and using amenability properties.
result Simplicial volume is finite for certain manifolds with amenable fundamental groups.
We construct a complete, embedded minimal surface in euclidean 3-space which has unbounded Gaussian curvature. It has infinite genus, infinitely many catenoidal type ends and one limit end.
The simplicial volume of non-R^3 contractible 3-manifolds is infinite.
problem Characterizing contractible 3-manifolds based on their simplicial volume.
method Analyzing the simplicial volume of contractible 3-manifolds and open 3-manifolds.
result The Euclidean space is the unique contractible 3-manifold with vanishing minimal volume.
New simplicial complex for infinite-type surfaces shows graph properties.
problem Characterizing infinite-type surfaces using graph theory.
method Constructing grand arc graph and analyzing its properties.
result Grand arc graph is infinite-diameter and δ-hyperbolic under certain conditions.
Study of harmonic functions on infinite penny graphs.
problem Characterizing harmonic functions on infinite penny graphs.
method Proving volume doubling and Poincaré inequalities, analyzing polynomial growth harmonic functions.
result Finite dimensional property of ancient solutions of the heat equation.
The topological type of a non-compact Riemann surface is determined by its ends space and the ends having infinite genus. In this paper for a non-compact Riemann Surface Sm,s with s ends and exactly m of them with infinite genus, such that m,s∈N and 1<m≤s, we give a precise description of t…
Embeddings of mapping tori for end-periodic graph maps are proven.
problem Embedding mapping tori of end-periodic graph maps into finite complexes.
method Flowline-preserving homotopy equivalence and π1-injective map. result Every mapping class of Γ arising from an end-periodic homotopy equivalence contains a representative whose mapping torus realizes such an embedding.