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. 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.
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.
Study harmonic functions on submanifolds and their cones.
problem Understanding harmonic functions on minimal submanifolds and their cylindrical cones.
method Using the method from [6, 7], analyzing polynomial growth.
result Dimensions of harmonic functions match if cone growth degree differs.
Minimal hypersurfaces with cylindrical tangent cones constructed and analyzed.
problem Constructing minimal hypersurfaces with specific geometric properties.
method Constructing minimal hypersurfaces with cylindrical tangent cones and proving unique continuation results.
result Existence and properties of minimal hypersurfaces with cylindrical tangent cones.
The study confirms a conjecture about critical points of smooth functions.
problem Understanding isolated critical points of smooth functions.
method Investigated cone-like, reasonable, and Rothe H hypothesis critical points.
result The conjecture holds true for certain critical points.
Extends a Liouville theorem for stable minimal hypersurfaces.
problem Stable minimal hypersurfaces in cylindrical cones.
method Analyzes the density and foliation of minimal hypersurfaces.
result Shows that stable minimal hypersurfaces are cylindrical.
Construct locally minimizing (1,2)-clusters with prescribed asymptotic geometry.
problem Minimizing clusters with prescribed asymptotic geometry.
method Develop a refined construction using the Hardt-Simon foliation.
result Produce a countably infinite family of distinct locally minimizing clusters asymptotic to a singular area-minimizing hypercone.
In this note, we prove that on a compact Kähler manifold X carrying a smooth divisor D such that KX+D is ample, the Kähler-Einstein cusp metric is the limit (in a strong sense) of the Kähler-Einstein conic metrics when the cone angle goes to 0. We further investigate the boundary behavior of those and prove th…
In this paper we show that any good toric contact manifold has well defined cylindrical contact homology and describe how it can be combinatorially computed from the associated moment cone. As an application we compute the cylindrical contact homology of a particularly nice family of examples that appear in the work of…
The minimal number of critical points is studied for smooth functions on closed manifolds.
problem Determining the minimal number of critical points for smooth functions on closed manifolds.
method Investigates cylindrical ball neighborhoods and exotic critical points, proving the conjecture for certain types of critical points.
result The minimal number of critical points is the same for smooth functions without exotic critical points on closed manifolds of dimension at least 6.
Proves uniqueness of blowups for forced mean curvature flow.
problem Proving uniqueness of blowups for forced mean curvature flow.
method Adapting methods from Euclidean space mean curvature flow to handle forcing term and blow-up limits.
result Uniqueness of tangent cones for forced mean curvature flow at self-shrinkers and cylindrical self-shrinkers.
We construct Lipschitz Q-valued functions which approximate carefully integral currents when their cylindrical excess is small and they are almost minimizing in a suitable sense. This result is used in two subsequent works to prove the discreteness of the singular set for the following three classes of 2-dimensiona…
Study area minimizing currents in Riemannian manifolds, proving unique structure and decay.
problem Area minimizing currents in Riemannian manifolds with moduli.
method Structural results and uniqueness theorem, inspired by Simon's techniques.
result Uniqueness and decay towards tangent cones for area minimizing currents.
The paper studies minimal graphs with bounded 2-dilation in Euclidean space.
problem Understanding minimal graphs with bounded 2-dilation in Euclidean space.
method Analyzing tangent cones and proving Neumann-Poincaré inequalities.
result Minimal graphs have multiplicity one tangent cones at infinity.
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…
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.
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.
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…
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 …
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.
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 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.
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.
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 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.
The paper studies cylindrical singularities in mean curvature flow and proves their local regularity.
problem Understanding the structure and regularity of cylindrical singular sets in mean curvature flow.
method Introduced a new L2-distance non-concentration property to prove the local regularity of singular sets. result Locally, cylindrical singular sets are contained in a k-dimensional C2,α-submanifold.