Rotational symmetry proven for special self-expanders near cylinders.
problem Proving symmetry of self-expanders near cylinders.
method Analyzing self-expanders to inverse mean curvature flow with specific end conditions.
result Proven rotational symmetry for self-expanders with cylindrical ends.
What are appropriate geometric conditions ensuring that a complete Riemannian 2-cylinder without conjugate points is flat? Examples with nonpositive curvature show that one has to assume that the ends of the cylinder open sublinearly. We show that sublinear growth of the ends is indeed sufficient if it is measured by t…
We prove the existence of a new class of constant mean curvature cylinders with an arbitrary number of umbilics by unitarizing the monodromy of Hill's equation.
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.
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.
The study proves uniqueness of certain types of solitons.
problem Proving uniqueness of asymptotically cylindrical shrinking Ricci solitons.
method Analyzing the behavior of solitons at spatial infinity and using order agreement conditions.
result Proven that such solitons are either isometric to the cylinder or its quotient.
The paper constructs isothermic surfaces using Ribaucour transformations.
problem Creating isothermic surfaces with specific properties.
method Applying Ribaucour transformations to the cylinder and obtaining families of complete isothermic surfaces.
result Obtained families of isothermic surfaces with unique properties (e.g., n-bubble surfaces, planar ends, no constant mean curvature).
Two families of genus one surfaces with H-curvature in H2imesR are constructed.
problem Constructing surfaces with positive constant mean curvature in H2imesR. method Conjugate construction.
result There is no Schoen-type theorem for immersed surfaces with positive constant mean curvature in H2imesR. Self-shrinkers in 3D have simple ends.
problem Understanding the structure of self-shrinkers.
method Analyzing asymptotic behavior of noncompact self-shrinkers.
result Each end of a self-shrinker is asymptotic to a cone or cylinder.
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.
The paper studies the structure at infinity of shrinking Ricci solitons with bounded curvature.
problem Understanding the structure at infinity of shrinking Ricci solitons with bounded curvature.
method Analyzes the limit behavior of complete gradient shrinking Ricci solitons and applies results to four-dimensional cases.
result The end of a shrinking Ricci soliton is asymptotic to a round cylinder at infinity.
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.
We construct higher genus Riemann's minimal surfaces properly embedded in the Euclidean space. To do that we glue end by end a Costa-Hoffman-Meeks examples to two halves genus zero Riemann's minimal surfaces. In first we need to perform a deformation of a Costa-Hoffman-Meeks example to prescribe the flux vector along t…
Study shows generic surfaces avoid complex flow patterns.
problem Understanding flow patterns of surfaces in 3D space.
method Analyzes mean curvature flow of closed surfaces in R3. result Non-cylindrical self-shrinkers cannot arise generically.
Asymptotically cylindrical Ricci-flat manifolds play a key role in constructing Topological Quantum Field Theories. It is particularly important to understand their behavior at the cylindrical ends and the natural restrictions on the geometry. In this paper we show that an orientable, connected, asymptotically cylindri…
Proves the only Type I κ-solution in 3D is the shrinking cylinder.
problem Classifying three-dimensional Type I κ-solutions to the Ricci flow.
method Analyzes the Ricci flow and proves the uniqueness of the solution.
result The only simply connected noncompact three-dimensional Type I κ-solution is the shrinking cylinder.
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.
This paper and its sequel prove a generalization of the usual gluing theorem for two index 1 pseudoholomorphic curves u_+ and u_- in the symplectization of a contact 3-manifold. We assume that for each embedded Reeb orbit gamma, the total multiplicity of the negative ends of u_+ at covers of gamma agrees with the total…
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.
This paper studies symplectic vortices on compact manifolds, proving their asymptotic behavior.
problem Analyzing the asymptotic behavior of finite energy symplectic vortices on compact manifolds.
method Proves the convergence of symplectic vortices to sectors of symplectic reduction at cylinder ends.
result Finite energy symplectic vortices exponentially converge to un/untwisted sectors of the symplectic reduction.
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.
Constructing translating solitons from Lagrangian Grim Reapers.
problem Creating Lagrangian translating solitons from intersections of Grim Reapers.
method Desingularizing intersections with special Lagrangian Lawlor necks.
result Constructing Lagrangian translating solitons with multiple ends and loops.
In this paper we present a new family of non-compact properly embedded, self-shrinking, asymptotically conical, positive mean curvature ends Σn⊆Rn+1 that are hypersurfaces of revolution with circular boundaries. These hypersurface families interpolate between the plane and half-cylinder in $\math…
These lectures are an introduction to formal semiclassical quantization of classical field theory. First we develop the Hamiltonian formalism for classical field theories on space time with boundary. It does not have to be a cylinder as in the usual Hamiltonian framework. Then we outline formal semiclassical quantizati…
We define an invariant ∇G(M) of pairs M,G, where M is a 3-manifold obtained by surgery on some framed link in the cylinder S×I, S is a connected surface with at least one boundary component, and G is a fatgraph spine of S. In effect, ∇G is the composition with the ιn maps of Le-Murakami-Ohtsu…
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…
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 …
Study finite curvature solutions on surfaces with nonnegative Gauss curvature.
problem Finite total curvature solutions of Liouville equation on surfaces with nonnegative Gauss curvature.
method Analyzes asymptotic behavior of solutions on complete surfaces.
result Two extremal cases identified: Euclidean plane or flat cylinder, with specific decay conditions.
Study on surfaces with constant anisotropic mean curvature in 3D space.
problem Characterizing surfaces with constant anisotropic mean curvature.
method Analyzing uniformly elliptic anisotropic functionals and proving properties of surfaces.
result Characterization of surfaces with constant anisotropic mean curvature.
New cohomology vanishing result for twisted cylinders.
problem Cohomology of twisted cylinders.
method Sobolev--Poincaré inequality methods.
result Vanishing result for Lq,p-cohomology. We prove that for a 7-dimensional manifold M with cylindrical ends the moduli space of exponentially asymptotically cylindrical torsion-free G_2 structures is a smooth manifold (if non-empty), and study some of its local properties. We also show that the holonomy of the induced metric of an exponentially asymptotically…
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 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.
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.
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.
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…
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 M2imesRn have zero curvature. 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.
A rather complete investigation of anisotropic Bessel potential, Besov, and Hölder spaces on cylinders over (possibly) noncompact Riemannian manifolds with boundary is carried out. The geometry of the underlying manifold near its 'ends' is determined by a singularity function which leads naturally to the study of weigh…
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.
We introduce constructions of exact Lagrangian cobordisms with cylindrical Legendrian ends and study their invariants which arise from Symplectic Field Theory. A pair (X,L) consisting of an exact symplectic manifold X and an exact Lagrangian cobordism L⊂X which agrees with cylinders over Legendrian links $…
Knots parametrized in cylinder coordinates by t -> (st, 3 + cos(nt), cos(mt + φ)) share properties of Lissajous and billiard knots in a cylinder. We use these 'billiard knots in a flat solid torus' to study two topics: when is Z(s,n,m) equal to Z(s,m,n)? And: why are the determinants of certain Lissajous and billiard k…
This paper explores how pairs of multicurves can be realized as cylinders on translation surfaces.
problem Understanding when pairs of multicurves can be realized as cylinders on translation surfaces.
method Surface topology and flat grafting deformation.
result Pairs of multicurves can be realized as cylinders on some translation surface.
New findings on κ-solutions with round cylinder as asymptotic shrinker.
problem Characterizing κ-solutions with specific asymptotic behavior. method Analysis of Ricci flow in dimensions n≥4. result Uniformly Positive Isoperimetric Constant (PIC) for κ-solutions. Study decomposes geometric surfaces, finding special curves.
problem Existence of certain affine diffeomorphisms on translation surfaces.
method Explicit cylinder decomposition on geometric surfaces.
result Special curves are obstructions to certain diffeomorphisms.
Study asymptotics of special curves on shrinking cylinders.
problem Asymptotic behavior of H−holomorphic curves on degenerating cylinders. method Analysis of H−holomorphic curves in symplectizations with harmonic perturbation. result Established precise asymptotic formulas for small area curves.
Round cylinders are rigid in Ricci shrinkers close to the standard product.
problem Proving rigidity of round cylinders in Ricci shrinkers.
method Proving isometry using pointed-Gromov-Hausdorff topology.
result Ricci shrinkers close to Sn−1imesR are isometric to Sn−1imesR. Let M be a closed, oriented, n-dimensional manifold. In this paper we give a Morse theoretic description of the string topology operations introduced by Chas and Sullivan, and extended by the first author, Jones, Godin, and others. We do this by studying maps from surfaces with cylindrical ends to M, such that on the c…