This paper studies mean curvature flows near cylindrical singularities.
problem Understanding the behavior of mean curvature flows near cylindrical singularities.
method Proved the rescaled flow converges to a graph over a cylinder, defined nondegeneracy, and showed properties of nondegenerate singularities.
result Nondegenerate cylindrical singularities are isolated, have a mean convex neighborhood, and are type-I.
We establish various existence and uniqueness results for the Yang-Mills flow on cylindrical end 4-manifolds. We also show long-time existence and infinite-time convergence under certain hypotheses on the underlying data.
We introduce cylindrical projections to simulate infinite-dimensional occupation flows of diffusions.
problem Computational intractability of infinite-dimensional occupation flows of diffusions.
method Introduce cylindrical projections to approximate the occupation flow via a finite-dimensional system.
result Strong convergence of cylindrical projections to the initial process with derived rates.
Paper proves strong uniqueness of cylindrical tangent flows near singularity in Ricci flow.
problem Proving strong uniqueness of cylindrical tangent flows near singularity in Ricci flow.
method Established Lojasiewicz inequality for pointed W-entropy under cylindrical geometry assumption. result Strong uniqueness of cylindrical tangent flows at first singular time of Ricci flow proved.
Eigenvalues of manifolds with cylindrical boundaries approximated by graph Laplacians.
problem Approximating eigenvalues of manifolds with cylindrical boundaries.
method Using truncated graph Laplacians constructed from (ε,ρ)-proximity graphs. result Eigenvalues of truncated graph Laplacians converge to Dirichlet eigenvalues of the Laplace-Beltrami operator.
Researchers found cylindrical steady gradient solitons in 3D.
problem Finding steady gradient solitons in 3D with specific symmetries.
method Constructed a two-parameter family of solitons with SO(2)imesR symmetry. result Found a family of solitons with asymptotic power-law or exponential decay.
We develop a new method to solve complex physics equations more accurately and efficiently.
problem Challenges in solving functional differential equations due to high computational costs and inaccurate approximations.
method Combining physics-informed neural networks (PINNs) with cylindrical approximation to handle functional derivatives.
result Our method achieves typical L1 relative error orders of PINNs of ∼10−3 on two FDEs. A cylindrical stretch line is a stretch line, in the sense of Thurston, whose horocyclic lamination is a weighted multicurve. In this paper, we show that two correctly parameterized cylindrical lines are parallel if and only if these lines converge towards the same point in Thurston's boundary of Teichmüller space.
Study proves uniqueness of asymptotic limits for specific manifolds.
problem Proving uniqueness of asymptotic limits for Ricci-flat manifolds with linear volume growth.
method Established using natural curvature and cross section assumptions.
result Uniqueness and exponential convergence rate for complete noncollapsed Ricci-flat manifolds with linear volume growth.
Proves convergence of mean curvature flow on cylinders with unique continuation.
problem Understanding the convergence and uniqueness of mean curvature flow on cylindrical surfaces.
method Proves convergence and provides unique continuation results for mean curvature flow on cylinders.
result Proves that rescaled mean curvature flow on cylinders converging super-exponentially must coincide with the cylinder itself.
In the present paper, we establish a gluing construction for the Nahm pole solutions to the Kapustin-Witten equations over manifolds with boundaries and cylindrical ends. Given two Nahm pole solutions with some convergence assumptions on the cylindrical ends, we prove that there exists an obstruction class for gluing t…
We consider Lipschitz-type backward stochastic differential equations (BSDEs) driven by cylindrical martingales on the space of continuous functions. We show the existence and uniqueness of the solution of such infinite-dimensional BSDEs and prove that the sequence of solutions of corresponding finite-dimensional BSDEs…
Sharp curvature pinching for mean curvature flow in spheres proved.
problem Proving sharp curvature pinching for mean curvature flow in spheres.
method Using blow-up arguments, codimension and cylindrical estimates, and rescaling.
result Smooth convergence to a totally geodesic limit in infinite time.
Study cylindrical symmetric Finsler metrics with vanishing Douglas curvature.
problem Understanding Finsler metrics with specific curvature properties.
method Analyzing cylindrically symmetric Finsler metrics and solving differential equations.
result Differential equations for cylindrically symmetric Finsler metrics with vanishing Douglas curvature.
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…
New steady Kähler-Ricci solitons found on orbifolds.
problem Constructing steady Kähler-Ricci solitons on orbifolds.
method Using asymptotically cylindrical structures and orbifold actions.
result Found new examples of gradient steady Kähler-Ricci solitons converging to a Ricci-flat cylinder.
Study resolves flow through cylindrical singularities, proving nonfattening.
problem Analyzing free boundary flow through cylindrical singularities.
method Foundational results for free boundary Brakke flows and classification of ancient flows.
result Proves all cylindrical singularities have a mean-convex neighborhood, leading to well-posed flow.
We study gluings of asymptotically cylindrical special Lagrangian submanifolds in asymptotically cylindrical Calabi--Yau manifolds. We prove both that there is a well-defined gluing map, and, after reviewing the deformation theory for special Lagrangians, prove that this gluing map defines a local diffeomorphism from m…
Study cylindrical symmetric Finsler metrics that are projectively flat.
problem Characterize Finsler metrics that are projectively flat.
method Solve the system of differential equations for cylindrical symmetric Finsler metrics.
result Provide a family of solutions for the projectively flat Finsler metrics.
In an earlier paper, we proved that, under certain hypotheses, the moduli space of an asymptotically cylindrical special Lagrangian submanifold with fixed boundary of an asymptotically cylindrical Calabi-Yau 3-fold is a smooth manifold. Here we prove the analogous result for an asymptotically cylindrical special Lagran…
Let M be a complete Ricci-flat Kahler manifold with one end and assume that this end converges at an exponential rate to [0,∞)×X for some compact connected Ricci-flat manifold X. We begin by proving general structure theorems for M; in particular we show that there is no loss of generality in assumi…
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.
Study on parabolic points and cylindrical surfaces in Euclidean 3-space.
problem Characterizing parabolic points and their geometric properties.
method Introducing contact cylindrical surfaces and analyzing their properties.
result Characterization of A-singularity through projections. We study the deformations of an asymptotically cylindrical Cayley submanifold inside an asymptotically cylindrical Spin(7)-manifold. We prove an index formula for the operator of Dirac type that arises as the linearisation of the deformation map and show that if the Spin(7)-structure is generic, then there are no obstr…
Proves uniqueness of cylindrical tangent flows in mean curvature flow.
problem Proving uniqueness of cylindrical singularity models in mean curvature flow.
method Inspired by Székelyhidi's approach, uses a different method to prove uniqueness.
result Proves uniqueness of cylindrical tangent flows.
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.
We study a class of asymptotically cylindrical Ricci-flat Kähler metrics arising on quasiprojective manifolds. Using the Calabi--Yau geometry and analysis and the Kodaira--Kuranishi--Spencer theory and building up on results of N.Koiso for the case of compact manifolds, we show that under rather general hypotheses any …
Unique cylindrical tangent cone for Simons' hypersurface found.
problem Uniqueness of cylindrical tangent cones for area-minimizing hypersurfaces.
method Developed a new Lojasiewicz inequality for non-isolated singularities.
result Cylindrical tangent cone for Simons' hypersurface is unique.
Uniqueness proven for cylindrical tangent cones in high dimensions.
problem Proving uniqueness of cylindrical tangent cones in high dimensions.
method Analyzing area-minimizing hypersurfaces in R^9.
result Uniqueness of cylindrical tangent cones Cp,qimesR in R9. For curves of prescribed length embedded into the unit disc in two dimensions, we obtain scaling results for the minimal elastic energy as the length just exceeds 2π and in the large length limit. In the small excess length case, we prove convergence to a fourth order obstacle type problem with integral constraint on…
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…
We develop some consequences of the connection between Calabi-Yau structures and torsion-free G2 structures on compact and asymptotically cylindrical six- and seven-dimensional manifolds. Firstly, we improve the known proof that matching asymptotically cylindrical Calabi-Yau threefolds can be glued. Secondly, we giv…
New special Lagrangian submanifolds with cylindrical tangent cones are constructed.
problem Constructing special Lagrangian submanifolds with specific geometric properties.
method Constructing examples in a neighborhood of the origin with an isolated singularity and cylindrical tangent cone.
result Existence of special Lagrangian submanifolds with cylindrical tangent cones, including examples with transverse planes.
New cylindrical solutions found for Grushin-type problem.
problem Critical Grushin-type problem on CR sphere.
method Local Pohozaev identities for non-degeneracy, Lyapunov-Schmidt reduction for solutions.
result New type of multi-bubbling cylindrical solutions constructed.
Develops a new method for financial term structure modeling.
problem Analyzing financial term structures with discontinuities.
method Cylindrical stochastic integration approach.
result Establishes a Heath-Jarrow-Morton framework.
We study a particular class of open manifolds. In the category of Riemannian manifolds these are complete manifolds with cylindrical ends. We give a natural setting for the conformal geometry on such manifolds including an appropriate notion of the cylindrical Yamabe constant/invariant. This leads to a corresponding ve…
Cylindrical contact homology computed for links of simple singularities.
problem Computing cylindrical contact homology for links of simple singularities.
method Perturbing degenerate contact form on S3/G with an invariant Morse function, achieving nondegeneracy up to an action threshold. Recovers cylindrical contact homology via direct limit of action filtered homology groups. result Ranks of cylindrical contact homology groups are given in terms of ∣extConj(G)∣, demonstrating a form of the McKay correspondence. 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.
In this paper we prove that any asymptotically cylindrical gradient shrinking Ricci soliton is isometric to a cylinder.
We construct solutions of the vacuum vector constraint equations on manifolds with cylindrical ends.
Researchers prove uniqueness of certain cylindrical tangent cones for special Lagrangians.
problem Proving uniqueness of cylindrical tangent cones for special Lagrangians.
method Analyzing exact special Lagrangian submanifolds with multiplicity one and cylindrical tangent cones.
result The cylindrical tangent cones are unique under specific conditions.
Resolves conjecture on cylindrical mean curvature flows in all dimensions.
problem Mean Convex Neighborhood Conjecture for cylindrical singularities.
method Complete classification of ancient, asymptotically cylindrical flows; refined asymptotic analysis; leading mode condition; induction over thresholds.
result Establishes mean-convex neighborhood for cylindrical singularities; provides local models and canonical families.
We construct examples of exponentially asymptotically cylindrical Riemannian 7-manifolds with holonomy group equal to G_2. To our knowledge, these are the first such examples. We also obtain exponentially asymptotically cylindrical coassociative calibrated submanifolds. Finally, we apply our results to show that one of…
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.
The paper studies cylindrical handlebody-knots of genus two with unique unknotting annuli and finds trivial symmetry groups.
problem Understanding the topology and symmetry of cylindrical handlebody-knots of genus two.
method Analysis of Thurston's hyperbolization theorem and investigation of unknotting annuli.
result The symmetry group is trivial if the unknotting annulus is unique and of type 2. Study proves uniqueness and rigidity of cylindrical self-shrinkers using Łojasiewicz inequalities.
problem Uniqueness and rigidity of cylindrical self-shrinkers in mean curvature flow.
method Direct perturbative analysis of the shrinker mean curvature and Łojasiewicz inequalities.
result Uniqueness and rigidity of cylindrical self-shrinkers, including round cylinders and cylinders over Abresch-Langer curves.
Study on cylindrical handlebody-knots with symmetry and rigidity properties.
problem Characterizing symmetry groups of cylindrical handlebody-knots of genus two.
method Classification of essential annuli and analysis of symmetry groups based on Koda-Ozawa theorem.
result Most exteriors of genus two cylindrical handlebody-knots contain no essential disks or tori, and type 3-3 annuli are often unique up to isotopy. Defines new Roe algebras for cylindrical spaces, solving metric curvature problems.
problem Existence and classification of metrics with positive scalar curvature on spaces with cylindrical ends.
method Variant of Roe algebras for cylindrical spaces, relating to relative higher index theory.
result Defines higher rho-invariants and provides a concise proof of a related result.