Finite number of eigenvalues found in cylindrical surface.
problem Finding eigenvalues in manifolds with cylindrical ends.
method Constructing a surface with a cylindrical end attached to a hyperbolic torus.
result First example of a finite and nonzero number of embedded eigenvalues in a cylindrical manifold.
Bounds on minimal surfaces' ends and stability index.
problem Understanding the topology and stability of minimal surfaces.
method Proving bounds on the number of ends and stability index of minimal surfaces.
result Minimal surfaces of genus g have a finite number of ends and a bounded stability index. Finite graphs with specific curvature have limited harmonic functions and ends.
problem Graphs with nonnegative curvature outside a finite subset.
method Introducing discrete Gromov-Hausdorff convergence to study bounded harmonic functions.
result The space of bounded harmonic functions is finite dimensional, and the number of non-parabolic ends is finite.
Finite genus embedded minimal surfaces have limited limit ends.
problem Characterizing limit ends of embedded minimal surfaces.
method Proving properties of limit ends and using them to deduce surface characteristics.
result Embedded minimal surfaces with finite genus have at most two limit ends.
We prove a version of the well-known Denjoy-Ahlfors theorem about the number of asymptotic values of an entire function for properly immersed minimal surfaces of arbitrary codimension in R^N. The finiteness of the number of ends is proved for minimal submanifolds with finite projective volume. We show, as a corollary, …
Every weak Perron number is realized as a stretch factor of a homeomorphism on a surface.
problem Finding stretch factors for weak Perron numbers.
method Constructing an end-periodic homeomorphism on a surface.
result Every weak Perron number is an end-periodic stretch factor.
Develops theory linking Schrödinger equations to manifold ends, proving finiteness.
problem Understanding the number of ends in Riemannian manifolds.
method Variant of Li-Tam theory, polynomial growth analysis, Sobolev inequality.
result Finiteness of manifold ends under scaling invariant Sobolev inequality.
Characterizes ends and coends of graph pairs using quasi-median graphs.
problem Understanding the number of ends and coends in graph pairs.
method Characterizes ends and coends using quasi-median graphs.
result Characterizes ends and coends of graph pairs (G,H) in terms of quasi-median graphs. The study counts ends on shrinkers using geometric covering methods.
problem Counting the number of ends on shrinkers.
method Geometric covering method to study the number of ends.
result Proves that the number of ends on any complete non-compact shrinker is at most polynomial growth with fixed degree.
Let Y be a noncompact rank one locally symmetric space of finite volume. Then Y has a finite number e(Y) > 0 of topological ends. In this paper, we show that for any natural number n, the Y with e(Y) \leq n that are arithmetic fall into finitely many commensurability classes. In particular, there is a constant c_n such…
Let f:A-->B be a covering map. We say A has e filtered ends with respect to f (or B) if for some filtration {K_n} of B by compact subsets, A - f^{-1}(K_n) "eventually" has e components. The main theorem states that if Y is a (suitable) free H-space, if K < H has infinite index, and if Y has a positive finite number of …
New metrics with constant Q-curvature created by gluing.
problem Creating metrics with constant Q-curvature on spheres with punctures.
method Gluing truncated known metrics together.
result Unmarked moduli space of solutions is nontrivial for at least four punctures.
The paper extends end concepts to arbitrary groups and spaces.
problem Extending end concepts to arbitrary groups and spaces.
method Description of maximal coarse compactification and geometric proof of end properties.
result Generalization of Stallings' theorem and definition of ends for coarse spaces.
Let M2 be a complete non compact orientable surface of non negative curvature. We prove in this paper some theorems involving parabolicity of minimal surfaces in M2×R. First, using a characterization of δ-parabolicity we prove that under additional conditions on M,…
We study the problem of existence of isoperimetric regions for large volumes, in C0-locally asymptotically Euclidean Riemannian manifolds with a finite number of C0-asymptotically Schwarzschild ends. Then we give a geometric characterization of these isoperimetric regions, extending previous results contained in …
Study on totally geodesic planes in hyperbolic 3-manifolds with specific properties.
problem Characterizing totally geodesic planes in hyperbolic 3-manifolds with degenerate ends.
method Ratner-type phenomenon and finite surface count approach.
result Closed minimal invariant subsets are either immersed totally geodesic surfaces or the entire manifold.
Study estimates self-shrinker index with conical ends, proving index bound.
problem Estimating the index of self-shrinkers with asymptotically conical ends.
method Constructing Gaussian Harmonic forms and extending index estimates.
result Proves Morse index of self-shrinkers is at least (2g+r-1)/3.
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…
Study classifies asymptotically flat manifolds with cone structure at infinity.
problem Classifying asymptotically flat manifolds with specific geometric properties.
method Analyzing the structure of manifolds with cone structure at infinity.
result Finite number of ends for asymptotically flat manifolds with cone structure at infinity.
The study explores ends in coarse homotopy of proper geodesic spaces.
problem Understanding ends in coarse homotopy of proper geodesic spaces.
method Recontextualizing ends as a functor and proving properties of coarse path components.
result Existence of a natural surjection from coarse path components to ends, not always an injection.
The study provides structure theorems for singular minimal laminations.
problem Understanding the structure of singular minimal laminations in 3D space.
method Local removable singularity theorem and local picture theorem applied to topology.
result Global structure theorems for certain singular minimal laminations.
Study on finite energy SU(2) monopoles on AC 3-manifolds, proving integrality of charge and curvature decay.
problem Understanding the asymptotic behavior of finite energy SU(2) monopoles on AC 3-manifolds.
method Analysis of critical points of the SU(2) Yang--Mills--Higgs energy on asymptotically conical 3-manifolds.
result Proves integrality of the monopole number and quadratic decay of curvature, among other findings.
We consider constant mean curvature surfaces of finite topology, properly embedded in three-space in the sense of Alexandrov. Such surfaces with three ends and genus zero were constructed and completely classified by the authors in arXiv:math.DG/0102183. Here we extend the arguments to the case of an arbitrary number o…
Open manifolds can be covered by Rm with finite or infinite degree.
problem Covering open manifolds with Rm. method Proving simple branched covers for open manifolds of dimensions 2 and 3, and investigating universal bases.
result Open manifolds can be covered by Rm with finite or infinite degree. 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.
Study of mapping class groups of infinite graphs, focusing on their finiteness and commensurability.
problem Understanding the finiteness properties and commensurability of mapping class groups of infinite graphs.
method Investigation of asymptotically rigid mapping class groups, construction of explicit presentations, and analysis of algebraic and geometric properties.
result Graph Houghton groups are not commensurable with other known Houghton-type groups, defining a new class of groups.
New theorem bounds group quotient size to subgroups index.
problem Understanding subgroup structure in hyperbolic groups.
method Proved quotient size bounds on Eilenberg-MacLane spaces.
result One-ended hyperbolic groups cannot have isomorphic finite-index subgroups.
Proves conditions for Willmore surfaces to have finite ends or finite total curvature.
problem Conditions for Willmore surfaces to have finite ends or finite total curvature.
method Analyzes scale-invariant second fundamental form near infinity.
result Proves conditions for Willmore surfaces to have finite ends or finite total curvature.
In this paper, we study the space of translational limits T(M) of a surface M properly embedded in R^3 with nonzero constant mean curvature and bounded second fundamental form. There is a natural map T which assigns to any surface M' in T(M), the set T(M') in T(M). Among various dynamics type results we prove that surf…
Study of minimal surfaces in 4D with specific ends.
problem Characterize minimal surfaces in R4 with specific ends. method Modification of Costa and Hoffman-Meeks method, generalized Weierstrass representation.
result Minimal surfaces with specific ends are J-holomorphic under certain conditions. Continuous epimorphisms between certain mapping class groups are induced by homeomorphisms.
problem Understanding continuous epimorphisms between specific mapping class groups.
method Analyzing subgroups of mapping class groups of infinite-genus 2-manifolds with no planar ends.
result Continuous epimorphisms are induced by homeomorphisms for the specified subgroups.
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.
The paper describes the K-theory of C∗-algebras of locally finite graphs.
problem Computing the K-theory of C∗-algebras of locally finite graphs. method Using a directed graph representation and Cuntz-Krieger algebra, the paper computes the K-theory of C∗(Γ). result The K-theory of C∗(Γ) is determined by the graph's genus, number of ends, and dead-ends. We show that Out(G) is residually finite if G is a one-ended group that is hyperbolic relative to virtually polycyclic subgroups. More generally, if G is one-ended and hyperbolic relative to proper residually finite subgroups, the group of outer automorphisms preserving the peripheral structure is residually finite. We…
We explicate a number of notions of algebraic laminations existing in the literature, particularly in the context of an exact sequence 1→H→G→Q→1 of hyperbolic groups. These laminations arise in different contexts: existence of Cannon-Thurston maps; closed geodesics exiting ends of manifolds; dual to …
A new layer learns abstract relations from graph structure using finite-state automata.
problem Learning abstract relations from graph structure for program analysis.
method Relaxing the problem into learning finite-state automata policies on a graph-based POMDP and training these policies using implicit differentiation.
result GFSA layer finds shortcuts in grid-world graphs and reproduces simple static analyses on Python programs.
Study relationships between submanifolds and ambient Kahler 4-manifolds' fundamental groups.
problem Relationships between submanifolds and fundamental groups of Kahler 4-manifolds.
method Analyzes fundamental groups of embedded Levi-flat or pseudoconvex submanifolds in Kahler 4-manifolds.
result Fundamental group of M4 determined by the fundamental group of compact embedded Levi-flat or pseudoconvex submanifolds. The paper studies minimal hypersurfaces in R^n with finite total curvature and proves index bounds.
problem Finite total curvature minimal hypersurfaces in R^n.
method Morse index and Jacobi operator analysis, rigidity case study, compactness and finiteness results.
result Linear lower bound on index and nullity for minimal hypersurfaces in R^n.
New proof of splitting theorem and finite ends of minimal hypersurfaces in nonnegative curvature manifolds.
problem Proving splitting theorem and finite ends of minimal hypersurfaces in nonnegative curvature manifolds.
method New proof of splitting theorem and construction of weighted minimizing geodesics at infinity.
result Minimal hypersurfaces with finite index in manifolds with nonnegative biRic curvature must have finite ends.
The plane and catenoid are the only capillary minimal surfaces outside a unit ball with one end and finite total curvature.
problem Characterizing capillary minimal surfaces in 3D space.
method Analytical proof using geometric properties and curvature constraints.
result The plane and catenoid are the only capillary minimal surfaces under specified conditions.
In this paper we prove that a complete minimal surface immersed in H^2xR, with finite total curvature and two ends, each one asymptotic to a vertical geodesic plane, must be a horizontal catenoid. Moreover, we give a geometric description of minimal ends of finite total curvature in H^2xR. We also prove that a minimal …
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.
A smooth end of a Bryant surface is a conformally immersed punctured disc of mean curvature 1 in hyperbolic space that extends smoothly through the ideal boundary. The Bryant representation of a smooth end is well defined on the punctured disc and has a pole at the puncture. The Willmore energy of compact Bryant surfac…
In this paper we shall establish that properly embedded constant mean curvature one surfaces in H^3 of finite topology are of finite total curvature and each end is regular. In particular, this implies the horosphere is the only simply connected such example, and the catenoid cousins the only annular examples of this n…
The goal of this paper is to study the geometry of cusped complex hyperbolic manifolds through their compactifications. We characterize toroidal compactifications with non-nef canonical divisor. We derive effective very ampleness results for toroidal compactifications of finite volume complex hyperbolic manifolds. We e…
In this paper we produce families of complete non compact Riemannian metrics with positive constant σk-curvature by performing the connected sum of a finite number of given n-dimensional Delaunay type solutions, provided 2≤2k<n. The problem is equivalent to solve a second order fully nonlinear elliptic eq…
Let G and F be finitely generated groups with infinitely many ends and let A and B be graph of groups decompositions of F and G such that all edge groups are finite and all vertex groups have at most one end. We show that G and F are quasi-isometric if and only if every one-ended vertex group of A is quasi-isometric to…
We prove that any complete, embedded minimal surface M with finite topology in a homogeneous three-manifold N has positive injectivity radius. When one relaxes the condition that N be homogeneous to that of being locally homogeneous, then we show that the closure of M has the structure of a minimal lamination o…