In this paper we show how the existence of a certain stable cylinder determines (locally) the ambient manifold where it is immersed. This cylinder has to verify a {\it bifurcation phenomena}, we make this explicit in the introduction. In particular, the existence of such a stable cylinder implies that the ambient manif…
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48 results for “cylinder manifolds”
Research shows curves in Walker 3-manifolds can lie in flat cylinders.
problem Understanding curves in Walker 3-manifolds.
method Showed curves lie in flat cylinders, constructed an example.
result Curves in Walker 3-manifolds can be contained in flat cylinders.
Study on instantons on cylinder manifolds with specific conditions.
problem Analyzing instantons on cylinder manifolds with non-zero smooth forms.
method Examined the instanton equation on cylinder manifolds with specific conditions.
result Instantons on good manifolds decay exponentially at the ends.
Proves a Minkowski inequality for star-shaped hypersurfaces in warped cylinders.
problem Proving a Minkowski inequality for specific types of hypersurfaces.
method Using weakly mean convex and star-shaped hypersurfaces in warped cylinders, and applying the inverse mean curvature flow.
result Sharp inequality holds for outward minimizing hypersurfaces in Schwarzschild and hyperbolic spaces.
Sutured manifolds defined by Gabai are useful in the geometrical study of knots and 3-dimensional manifolds. On the other hand, homology cylinders are in an important position in the recent theory of homology cobordisms of surfaces and finite-type invariants. We study a relationship between them by focusing on sutured …
Higher order higher spin operators are generalizations of -powers of the Dirac operator. In this paper, we study higher order higher spin operators defined on some conformally flat manifolds, namely cylinders and Hopf manifolds. We will also construct the kernels of these operators on these manifolds.
A homology cylinder over a compact manifold is a homology cobordism between two copies of the manifold together with a boundary parametrization. We study abelian quotients of the homology cobordism group of homology cylinders. For homology cylinders over general surfaces, it was shown by Cha, Friedl and Kim that their …
Ancient solutions to Ricci flow in 3D are mostly cylinders or solitons.
problem Understanding finite-time singularities in Ricci flow on compact 3-manifolds.
method Proved every noncompact ancient κ-solution in 3D is isometric to specific models.
result Ancient κ-solutions in 3D are either shrinking cylinders or the Bryant soliton.
Sharp inequalities link manifold's systole to curvature of boundary.
problem Finding relationships between manifold's systole and curvature.
method Proved inequalities relating homological systoles to scalar and mean curvature.
result Equality case implies universal cover is a cylinder.
New theorem splits 3-manifolds with non-negative curvature.
problem Splitting 3-manifolds with non-negative scalar curvature.
method Uses area-minimizing cylinders to prove flatness.
result 3-manifolds with non-negative scalar curvature are flat if they contain an area-minimizing cylinder.
Study geodesics on a special cylinder with arbitrary wind.
problem Global behavior of geodesics on a Randers metric cylinder.
method Solve Zermelo's navigation problem to define the Randers metric; analyze geodesics, conjugate, and cut loci.
result Characterize geodesics and their properties on the base manifold.
Study singularity formation in Ricci flow solutions.
problem Understanding singularity behavior in noncompact manifolds.
method Analyzing complete Ricci flow solutions.
result Evidence for stability of generalized cylinders as singularity models.
In this paper, we study a notion of hyperbolicity for hyperbolicity foliations with 1-dimensional parabolic leaves, namely the non-existence of holomorphic cylinders along the foliation - holomorphic maps from $\D^{n-1} \times \C$ to the manifold sending each $\{*\} \times \C$ to a leaf. We construct a tensor, that is …
Study strong maximum principles for mean curvature operators on subriemannian manifolds.
problem Investigate strong maximum principles for mean curvature operators on subriemannian manifolds.
method Analyze subriemannian manifolds including Heisenberg groups and cylinders, under Hormander type conditions.
result Show strong maximum principles for horizontal (p-) mean curvature operator and p-(sub)laplacian operator under certain conditions.
Center manifold analysis can be used in order to investigate the stability of the stationary solutions of various PDEs. This can be done by considering the PDE as an ODE between certain Banach spaces and linearising about the stationary solution. Here we investigate the volume preserving mean curvature flow using such …
We study constrained generalized Killing spinors over the metric cone and cylinder of a (pseudo-)Riemannian manifold, developing a toolkit which can be used to investigate certain problems arising in supersymmetric flux compactifications of supergravity theories. Using geometric algebra techniques, we give conceptually…
A splitting theorem for 3-manifold with nonnegative scalar curvature and mean-convex boundarymath.DG
New theorem on 3-manifolds with curvature and convex boundary.
problem Understanding 3-manifolds with specific curvature and boundary properties.
method Analyzes properties of Riemannian 3-manifolds with nonnegative scalar curvature and mean-convex boundary.
result Shows flatness of certain 3-manifolds containing specific geometric objects.
The paper characterizes D'Atri spaces using total scalar curvature of hemispheres.
problem Characterizing D'Atri spaces using geometric properties.
method Characterization of D'Atri spaces via total scalar curvature of geodesic hemispheres.
result A 3D Riemannian manifold is a D'Atri space if and only if the total scalar curvature of tubes about geodesic segments holds.
Paper proves rigidity of certain Ricci shrinkers.
problem Rigidity of Ricci shrinkers in specific spaces.
method Quantitative characterization, rigidity inequality, contraction and extension.
result Uniqueness of tangent flow for compact Ricci flows.
We consider a homological enlargement of the mapping class group, defined by homology cylinders over a closed oriented surface (up to homology cobordism). These are important model objects in the recent Goussarov-Habiro theory of finite-type invariants of 3-manifolds. We study the structure of this group from several d…
Study of degenerating maps to Riemannian manifolds, proving asymptotic limits and existence of minimal cylinders.
problem Blow-up analysis and behavior of maps from degenerating surfaces to compact manifolds.
method Blow-up analysis, generalized energy identities, neck asymptotic limits, evolution system for minimal cylinders.
result Asymptotic limits of necks are geodesics or geodesic-like curves, confirming conjectures.
In the present paper we give a geometric proof for the existence of cylinders with constant mean curvature in certain simply connected homogeneous three-manifolds diffeomorphic to , which always admit a Lie group structure. Here, denotes the critical value for which constant mean curva…
Billiard knots in a cylindermath.GT
We define cylinder knots as billiard knots in a cylinder. We present a necessary condition for cylinder knots: after dividing cylinder knots by possible rotational symmetries we obtain ribbon knots. We obtain an upper bound for the number of cylinder knots with two fixed parameters (out of three). In addition we prove …
Torsion elements in the associated graded of the -filtration of the monoid of homology cylindersmath.GT
This paper detects torsion elements in homology cylinder monoids.
problem Detecting torsion elements in the associated graded of the Y-filtration of homology cylinders.
method Introduced a homomorphism induced by the LMO functor to detect torsion elements.
result Every non-trivial torsion element in has order 3.
Classifies geodesic-preserving bijections in Thurston geometries.
problem Identifying bijections that preserve geodesics in different geometries.
method Comprehensive classification and proof for various geometries.
result Complete classification of geodesic-preserving bijections in Thurston geometries.
Uniqueness results and enclosure properties for hypersurfaces with boundary in weighted cylindersmath.DG
The study characterizes hypersurfaces in weighted cylinders and generalizes confinement properties.
problem Characterizing hypersurfaces in weighted Riemannian products.
method Analyzing parabolic hypersurfaces with boundary in weighted cylinders.
result Generalized confinement properties of hypersurfaces in weighted cylinders.
Paper proves genus of surfaces decreases in mean curvature flow.
problem Understanding the evolution of surfaces under mean curvature flow.
method Analyzes mean curvature flow of compact surfaces in 3-manifolds with specific curvature conditions.
result Genus of the regular set decreases over time.
Study finds solutions to inequality decay to zero on warped cylinders.
problem Analyzing solutions to a specific inequality on warped cylindrical ends.
method New Carleman estimate for independent interest.
result Solutions decay to zero along warped cylindrical ends.
Geodesics in positive Lagrangian spaces map to special Lagrangian cylinders.
problem Understanding geodesics in spaces of positive Lagrangian submanifolds.
method Cylindrical transform of geodesics, solving elliptic PDEs.
result Geodesics in positive Lagrangian spaces correspond to one-parameter families of special Lagrangian cylinders.
For a compact manifold, which has a part isometric to a cylinder of finite length, we consider an adiabatic limit procedure, in which the length of the cylinder tends to infinity. We study the asymptotic of the spectrum of Hodge-Laplacian and the asymptotic of the -metric on de Rham cohomology. As an application, …
New subdomains found on cylinder and sphere with nonconstant curvatures.
problem Existence of subdomains with specific curvature properties.
method Functional analytic approach to overcome regularity issues.
result First examples of nonhomogeneous, nonisoparametric subdomains.
New cohomology vanishing result for twisted cylinders.
problem Cohomology of twisted cylinders.
method Sobolev--Poincaré inequality methods.
result Vanishing result for -cohomology.
The fundamental properties of -holomorphic maps depend on two inequalities: The gradient inequality gives a pointwise bound on the differential of a -holomorphic map in terms of its energy. The cylinder inequality stipulates and quantifies the exponential decay of energy along cylinders of small total energy. We …
We study algebraic structures of certain submonoids of the monoid of homology cylinders over a surface and the homology cobordism groups, using Reidemeister torsion with non-commutative coefficients. The submonoids consist of ones whose natural inclusion maps from the boundary surfaces induce isomorphisms on higher sol…
Minimal cylinders in Heisenberg group characterized using loop group method.
problem Characterizing minimal cylinders in the Heisenberg group.
method Generalized Weierstrass type representation and loop group method.
result Characterization of all non-vertical minimal cylinders in terms of pairs of closed plane curves.
The Cylinder Theorem in math.DG
The paper proves cylinders in hyperbolic 3-space have zero curvature.
problem Characterizing surfaces with zero curvature in hyperbolic 3-space.
method Analyzes cylinders and applies Gauss and extrinsic curvature conditions.
result Proves complete surfaces with zero curvature are cylinders.
The paper proves conditions for positive scalar curvature and Yamabe constant on noncompact cylinders.
problem Conditions for positive scalar curvature and Yamabe constant on noncompact cylinders.
method Analyzes complete metrics with positive scalar curvature and Yamabe constant on noncompact cylinders.
result Positive scalar curvature and Yamabe constant conditions are satisfied under specific geometric and conformal class constraints.
Holomorphic cylinders converge to disks joined by flow lines.
problem Convergence of perturbed holomorphic cylinders.
method Exponential estimates and flow line computation.
result Holomorphic cylinders converge to two disks joined by a flow line.
Lipshitz, Ozsváth and Thurston defined a bordered Heegaard Floer invariant CFDA for 3-manifolds with two boundary components, including mapping cylinders for surface diffeomorphisms. We define a related invariant for certain 4-dimensional cobordisms with corners, by associating a morphism F from CFDA(f) to CFDA(g) to e…
We prove the existence of a countable family of Delaunay type domains Ω_j in M^n x R, where M^n is the Riemannian manifold S^n or H^n and n is at least 2, bifurcating from the cylinder B^n x R (where B^n is a geodesic ball of radius 1 in M^n) for which the first eigenfunction of the Laplace-Beltrami operator with zero …
The paper proves properties of open manifolds with positive isotropic curvature.
problem Characterizing complete noncompact manifolds with positive isotropic curvature.
method Ricci flow with surgery on open orbifolds with isolated singularities.
result Open manifolds are diffeomorphic to a connected sum of spherical manifolds and cylinders.
Two ancient solutions to Gauss curvature flow are identified for cylinders.
problem Classifying ancient solutions to Gauss curvature flow in cylinders.
method Assumption of cylinder cross-section bounded convexity, analysis of asymptotic behavior.
result Only two ancient solutions identified: translating soliton and compact oval solution.
The paper develops a method to map knots in a cylinder to virtual-flat knots.
problem How to map knots in a cylinder to virtual knots.
method Construct a diagram on a cylinder with invisible crossings, then pull back invariants.
result Developed a method to map knots in a cylinder to virtual-flat knots.
The study classifies quasi-Einstein manifolds with boundary.
problem Classifying quasi-Einstein manifolds with boundary.
method Analyzing scalar curvature and isometric properties.
result Compact quasi-Einstein manifolds with boundary are rigid.
On the Cylinder Theorem in math.DG
Cylinders in warped product spaces have zero curvature.
problem Characterizing cylinders in warped product spaces.
method Proving cylinders have zero extrinsic and intrinsic curvatures.
result Cylinders in have zero curvature.
This paper decomposes Hodge cohomology of manifold cylinders over graphs.
problem Decomposing Hodge cohomology of manifolds with embedded cylinders.
method Develops adiabatic decomposition using combinatorial and geometric structures.
result Generalizes Cappell-Lee-Miller splicing map to finite number of edges.
The paper studies cylinder curves in flat metrics with q > 2.
problem Characterizing behaviors of embedded cylinder curves in flat metrics with q > 2.
method Constructing examples and proving properties of cylinder curves.
result Embedded cylinder curves form a finite diameter subset of the curve complex when the surface is fully punctured and the metric has a specific form.
The paper examines the stability of Killing cylinders in hyperbolic space.
problem Stability of Killing cylinders in hyperbolic space.
method Explicit computation of Morse index for Jacobi operator on various support surfaces.
result Delaunay surfaces can be bifurcated from Killing cylinders supported on geodesic planes.