Study of non-metrizable manifolds' ends, generalizing Nyikos's theorem.
problem Characterizing non-metrizable surfaces with ends.
method Introducing short and long ends, developing theory of spaces of ends.
result Characterization of surfaces as metrizable manifolds plus long pipes.
Survey on manifold ends with new heat kernel estimates.
problem Analyzing geometric properties on manifolds with ends.
method Constructing manifolds with ends and analyzing their heat kernel estimates.
result Found manifolds with ends that have different heat kernel estimates.
Optimal Poincaré constant estimates on manifolds with ends.
problem Estimating the Poincaré constant on manifolds with ends.
method Heat kernel estimates extended to manifolds with ends, focusing on central balls.
result The Poincaré constant is determined by the second largest end.
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.
A manifold's discreteness is tied to its number of ends.
problem Understanding the discreteness of manifolds.
method Analyzing Freudenthal compactification to count ends.
result Strongly discrete manifolds have exactly one end.
Free groups can be end homogeneity groups of 3-manifolds.
problem Tackling the possibility of free groups as end homogeneity groups of 3-manifolds.
method Constructing specific 3-manifolds with end homogeneity groups isomorphic to free groups.
result For every finitely generated free group, there exists an irreducible open 3-manifold with that group as its end homogeneity group.
The study finds monotonic properties of harmonic functions on 3-manifolds with a flat end.
problem Understanding harmonic functions on 3-manifolds with specific ends.
method Derives monotonic properties of positive harmonic functions on 3-manifolds with nonnegative scalar curvature and asymptotically flat ends.
result Rigidity characterization of spatial Schwarzschild manifolds with two ends.
Researchers create a parametrix for resolvents on manifolds with ends.
problem Essential self-adjointness of elliptic symmetric differential operators on manifolds with ends.
method Introduced semiclasical pseudodifferential operators compatible with the end structure.
result Essential self-adjointness of elliptic symmetric differential operators proved.
New positive mass theorems for ALH manifolds with toroidal ends.
problem Proving positive mass theorems for asymptotically locally hyperbolic manifolds.
method Utilizes properties of marginally outer trapped surfaces and a new technique involving μ-bubbles.
result Obtained new positive mass theorems for asymptotically locally hyperbolic manifolds without boundary.
This paper proves unique determination of certain non-quasi-Fuchsian manifolds by their end structure and bending lamination.
problem Unique determination of non-quasi-Fuchsian manifolds by their end structure and bending lamination.
method Analysis of the end structure (parabolic locus, ending laminations, conformal structures) and bending lamination.
result Non-quasi-Fuchsian manifolds are uniquely determined by their end structure and bending lamination.
Extends ASD connection existence to 4-manifolds with cylindrical ends.
problem Existence of ASD connections on 4-manifolds with specific ends.
method Extends gluing theorems to cylindrical ends, using mASD connections.
result Establishes existence of ASD connections on 4-manifolds with cylindrical ends.
Researchers found multiple ways to end-sum 4-manifolds, contradicting a previous conjecture.
problem Nonuniqueness of end-sums in 4-manifolds.
method Explicit examples and detailed discussion of end-cohomology algebra.
result Uncountably many distinct proper homotopy types from end-sums.
Study finds solutions for complex problems on non-compact manifolds.
problem Solving fully nonlinear Yamabe-type problems on non-compact manifolds.
method Existence results for a class of problems, considering both positive and negative cases.
result Explicit examples of manifolds satisfying the hypotheses of the theorems.
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 end sum for surfaces and prove uniqueness results.
problem Uniqueness of end sum for surfaces and related manifolds.
method Analyzing end sum and adding a 1-handle at infinity for surfaces.
result The end sum of two surfaces with compact boundary is uniquely determined by the chosen ends.
We study the spectral theory of asymptotically hyperbolic manifolds with ends of warped product type. Our main result is an upper bound on the resonance counting function with a geometric constant expressed in terms of the respective Weyl constants for the core of the manifold and the base manifold defining the ends.
Researchers prove positive mass theorem for manifolds with arbitrary ends.
problem Proving the positive mass theorem for manifolds with non-compact ends.
method Developed techniques to handle non-compact singular sets and used Wloc1,p metrics. result Established positive mass theorem for C0 arbitrary ends with Wloc1,p metrics. New 3D shapes can't be split into torus pieces.
problem 3D shapes without torus decompositions.
method Recursive definition from compact 3-manifolds.
result Examples of 3D shapes failing torus decomposition.
We show 10/8-type inequalities for some end-periodic 4-manifolds which have positive scalar curvature metrics on the ends. As an application, we construct a new family of closed 4-manifolds which do not admit positive scalar curvature metrics.
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.
Proves mass theorem for manifolds with arbitrary ends.
problem Proving the positive mass theorem for manifolds with various ends.
method Quantitative analysis of scalar curvature on manifolds with arbitrary ends.
result Proves the positive mass theorem for a wide class of manifolds.
Real projective structures on n-orbifolds are useful in understanding the space of representations of discrete groups into SL(n+1,R) or PGL(n+1,R). A recent work shows that many hyperbolic manifolds deform to manifolds with such structures not projectively equivalent to the original ones. The …
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.
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.
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.
This paper studies Dirac operators on end-periodic spin manifolds of dimension at least 4. We provide a sufficient condition for such an operator to be Fredholm for a generic end-periodic metric; this condition is shown to be necessary in dimension 4. We make use of end-periodic Dirac operators to give an analytical in…
Proves positive mass theorem for specific manifold types.
problem Positive mass theorem for manifolds with arbitrary ends.
method Proof for asymptotically flat and Euclidean manifolds.
result Validates positive mass theorem in new manifold types.
Study examines Lp-boundedness of Hodge projection on manifolds with ends.
problem Understanding Lp-boundedness of Hodge projection on manifolds with ends. method Investigates the relationship between Hodge projection, Riesz transform, and bounded harmonic functions.
result Connects Lp-boundedness of Hodge projection to the structure of L2 harmonic one-forms and bounded harmonic functions. 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.
A local deformation property for uniform embeddings in metric manifolds (LD) is formulated and its behaviour is studied in a formal view point. It is shown that any metric manifold with a geometric group action, typical metric spaces (Euclidean space, hyperbolic space and cylinders) and for κ\leq 0 the κ-cone ends over…
Real projective structures on n-orbifolds are useful in understanding the space of representations of discrete groups into SL(n+1,R) or PGL(n+1,R). A recent work shows that many hyperbolic manifolds deform to manifolds with such structures not projectively equivalent to the o…
We study Fredholm properties and index formulas for Dirac operators over complete Riemannian manifolds with straight ends. An important class of examples of such manifolds are complete Riemannian manifolds with pinched negative sectional curvature and finite volume.
Real projective structures on n-orbifolds are useful in understanding the space of representations of discrete groups into SL(n+1,R) or PGL(n+1,R). A recent work shows that many hyperbolic manifolds deform to manifolds with such structures not projectively equivalent to the o…
Study adiabatic limits of Calderon projector on manifolds with cylindrical ends.
problem Constructing Calderon projector in a specific geometric setting.
method Using resolvent estimates and b-calculus framework.
result Extend a result on adiabatic limits of Cauchy data spaces.
The paper solves the Dirichlet problem at infinity for certain negatively curved 3-manifolds.
problem Solving the Dirichlet problem at infinity for negatively curved 3-manifolds with expansive ends.
method Based on a result that does not require explicit curvature assumptions, the paper presents an example of a metric on an end with indefinite curvature for which the Dirichlet Problem at Infinity is solvable.
result The Dirichlet problem at infinity is solvable for certain negatively curved 3-manifolds with expansive ends.
For every finitely generated abelian group G, we construct an irreducible open 3-manifold MG whose end set is homeomorphic to a Cantor set and with end homogeneity group of MG isomorphic to G. The end homogeneity group is the group of self-homeomorphisms of the end set that extend to homeomorphisms of the 3-m…
The paper proves the existence of area-minimizing hypersurfaces in AF manifolds of higher dimensions.
problem Existence of area-minimizing hypersurfaces in AF manifolds with arbitrary dimension and ends.
method Positive mass theorem for AF manifolds with arbitrary ends and global behavior for hypersurfaces in AF manifolds of dimension ≤ 8.
result Existence and behavior of area-minimizing hypersurfaces in AF manifolds of higher dimensions.
We show that for every sequence (ni), where each ni is either an integer greater than 1 or is ∞, there exists a simply connected open 3-manifold M with a countable dense set of ends {ei} so that, for every i, the genus of end ei is equal to ni. In addition, the genus of the ends not in the d…
The study analyzes spectral asymmetry and index theory on manifolds with generalized hyperbolic cusps.
problem Analyzing spectral asymmetry and index theory on manifolds with generalized hyperbolic cusps.
method Equivariant index theorem for Dirac operators on manifolds with φ-cusps under conditions on φ. result The cusp contribution is zero if the spectrum of the relevant Dirac operator on a hypersurface is symmetric around zero.
We use Furuta's result, usually referred to as ``10/8-conjecture'', to show that for any compact 3-manifold M the open manifold $M\times\r$ has infinitely many different smooth structures. Another consequence of Furuta's result is existence of infinitely many smooth structures on open topological 4-manifolds with a t…
Motivated by a recent groundbreaking work of Ontaneda, we describe a sizable class of closed manifolds such that the product of each manifold in the class with the real line admits a complete metric of bounded negative sectional curvature which is an exponentially warped near one end and has finite volume near the othe…
We give a simple construction of new, complete, finite volume manifolds M of bounded, nonpositive curvature. These manifolds have ends that look like a mixture of locally symmetric ends of different ranks and their fundamental groups are not duality groups.
This paper uses Brin and Thickstun's theory of end reductions of non-compact 3-manifolds to study groups of covering translations of irreducible contractible open 3-manifolds W which are not homeomorphic to R^3. We associate to W an object S(W) called the simplicial complex of minimal R^2-irreducible end reductions of …
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.
Real projective structures on n-orbifolds are useful in understanding the space of representations of discrete groups into SL(n+1,R) or PGL(n+1,R). A recent work shows that many hyperbolic manifolds deform to manifolds with such structures not projectively equivalent to the o…
Heat kernel estimates on manifolds with mixed boundary conditions.
problem Estimating heat kernels on manifolds with ends and mixed boundary conditions.
method Global harmonic function construction and h-transform technique. result Two-sided heat kernel estimates for Riemannian manifolds with mixed boundary 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…
We extend the Atiyah, Patodi, and Singer index theorem for first order differential operators from the context of manifolds with cylindrical ends to manifolds with periodic ends. This theorem provides a natural complement to Taubes' Fredholm theory for general end-periodic operators. Our index theorem is expressed in t…