Paper solves long neck problem on odd-dimensional spin manifolds.
problem Long neck problem on odd-dimensional spin manifolds.
method Spectral flow of Callias operators.
result Complete answer to Gromov's long neck problem.
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48 results for “long neck problem”
A note on the long neck principle and spectral width inequality of geodesic collar neighborhoodsmath.DG
Derives generalizations of the long neck principle and spectral width inequality.
problem Understanding the spectral width of geodesic collar neighborhoods.
method Spinorial Callias operator approach and relative Gromov-Lawson pair.
result Generalizations of the long neck principle and spectral width inequality.
Proves principles and estimates for initial data sets in Einstein equations.
problem Understanding initial data sets in Einstein equations.
method Spinorial Callias operator approach.
result Proves long neck principle and width estimates.
We study moduli spaces of Seiberg-Witten monopoles over spin^c Riemannian 4-manifolds with long necks and/or tubular ends. This first part discusses compactness, exponential decay, and transversality. As applications we prove two vanishing theorems for Seiberg-Witten invariants.
We establish a gluing theorem for monopoles over 4--manifolds containing long necks. The theorem is stated in terms of an ungluing map defined explicitly in terms of data that appear naturally in applications. Orientations of moduli spaces are handled using Benevieri--Furi's concept of orientations of Fredholm operator…
Metastability of lipid necks via geometric trialitycond-mat.soft
Lipid necks, large curvature bridges, are shown to be metastable.
problem Understanding the energetically prohibitive yet ubiquitous lipid necks in cell membranes.
method Geometric triality approach to demonstrate metastability.
result Lipid necks can exist for finite but potentially long times without stabilizing mechanisms.
The paper constructs harmonic 1-forms on 3-manifolds with cylindrical necks.
problem Stabilizing Z/2-harmonic 1-forms on closed 3-manifolds.
method Explicit construction of harmonic 1-forms by modifying metrics near links.
result Construction of harmonic 1-forms degenerating to manifolds with cylindrical ends.
We establish that finite-time singularities do not occur in four-dimensional Yang-Mills flow, confirming the conjecture of Schlatter, Struwe, and Tahvildar-Zadeh. The proof relies on a weighted energy identity and sharp decay estimates in the neck region.
Paper proposes a smart neck-band for detecting neck postures using integrated kinematic and kinetic data.
problem Improper neck postures lead to musculoskeletal disorders requiring therapy and rehabilitation.
method Integrated use of kinematic and kinetic data with machine learning algorithms.
result 100% accuracy in predicting neck postures using the proposed platform.
Study on harmonic maps from surfaces with energy bounds and neck domains.
problem Behavior of harmonic maps with bounded energy on complex domains.
method Analysis of a sequence of harmonic maps in generalized neck domains.
result Upper bound of energy density and study of nullity and index limits.
The paper extends energy identities and neck existence for ε-harmonic maps.
problem Understanding the energy identity and neck formation for ε-harmonic maps.
method Finding analogues of energy identities and neck existence results for ε-harmonic maps.
result Specific quantities determine energy identity and neck formation for ε-harmonic maps.
Study neck pinches in Lagrangian flows, proving stability and introducing new singularities.
problem Understanding neck pinches in Lagrangian flows.
method Introduced nondegenerate neck pinch and teardrop singularities, proving stability and answering questions.
result Nondegenerate neck pinches are stable and can be perturbed to nondegenerate singularities.
Paper proves energy identity and no-neck property for special harmonic maps.
problem Analyzing special harmonic maps with homogeneous targets.
method Introduced equivariant embedding for -harmonic case.
result Energy identity and no-neck property established for - and -harmonic maps.
A multi-neck spacetime wormhole is constructed with a simple metric tensor.
problem Existence of multi-neck spacetime wormholes.
method Spherical inversion of a 3-torus to create a 3-neck spacetime wormhole.
result Exact solution of Einstein's field equations for a multi-neck spacetime wormhole.
We introduce Lagrangian mean curvature flow with boundary in Calabi--Yau manifolds by defining a natural mixed Dirichlet-Neumann boundary condition, and prove that under this flow, the Lagrangian condition is preserved. We also study in detail the flow of equivariant Lagrangian discs with boundary on the Lawlor neck an…
It is a fundamental open problem for the mean curvature flow, and in fact for many partial differential equations, whether or not all blowup limits are selfsimilar. In this short note, we prove that for the mean curvature flow of mean convex surfaces all limit flows are selfsimilar (static, shrinking or translating) if…
Explains examples of Lagrangian flow with circle symmetry.
problem Understanding Lagrangian flow with symmetry.
method Examining specific examples in .
result Shows various types of flow, including compact and non-compact.
In this paper we study motion of surfaces of revolution under the mean curvature flow. For an open set of initial conditions close to cylindrical surfaces we show that the solution forms a "neck" which pinches in a finite time at a single point. We also obtain a detailed description of the neck pinching process.
In this paper, we prove some refined estimate in the neck region when a sequence of harmonic maps from surfaces blow up. The new estimate allows us to see the shape of the center of the neck region. As an application, we prove an inequality about the nullity and index when blow-up occurs.
Study geodesics on neck-degenerate manifolds, focusing and winding behavior observed.
problem Geodesics behavior on neck-degenerate manifolds with cuspidal singularities.
method Detailed multiscale analysis, blow-up techniques.
result Geodesics exhibit focussing and winding behavior as the neck degenerates.
The shape equation and linking conditions for a vesicle with two-phase domains are derived. We refine the conjecture on the general neck condition for the limit shape of a budding vesicle proposed by Jülicher and Lipowsky [Phys. Rev. Lett. \textbf{70}, 2964 (1993); Phys. Rev. E \textbf{53}, 2670 (1996)], and then we us…
We prove the energy identity and the no neck property for a sequence of smooth extrinsic polyharmonic maps with bounded total energy.
We find calibrated submanifolds in neck manifolds. Particularly, we obtain a calibrated submanifold in the Lagrangian self-expander constructed by Joyce, Lee and Tsui.
The Mathematics of Tie Knotsmath.GT
Classifies 85 tie knots into mathematical categories.
problem Classifying and understanding the mathematical properties of tie knots.
method Formal language and sequence of moves to describe tie knots, classification based on knot theory.
result Proves that any tie knot is prime and alternating.
In this paper we are dealing with mean curvature flow with surgeries of two-convex hypersurfaces. The main focus is to expand on the discussion in Section of Mean Curvature Flow with Surgeries of Two-Convex Hypersurfaces by Huisken and Sinestrari. Firstly we wish to establish how the neck detection lemma allows us …
We will give a weak energy identity for Sacks-Uhlenbeck approximation of harmonic maps and calculate the length of the necks.
Study controls curvature in Ricci flows using necks.
problem Controlling curvature in Ricci flows.
method Introducing necks of maximal symmetry and decomposing curvature into uniform bounds.
result Established -bounds on Riemann curvature tensor.
Stability of a special spacetime solution is proven under certain symmetries.
problem Understanding the long-time behavior of cosmological solutions with symmetries.
method Proves stability of double-cusp spacetime solution under small T2-symmetry-preserving perturbations.
result Double-cusp solution is stable under small T2-symmetry-preserving perturbations.
Mean curvature flow shows singularities on smooth surfaces.
problem Understanding singularities in mean curvature flow.
method Analyzing spherical or nondegenerate neck pinches.
result First singular time has isolated singularities.
The paper proves the existence of maxfaces with multiple swallowtails and planar ends.
problem Existence of maxfaces with specific geometric properties.
method Analyzes 1-parameter infinite genus families of maxfaces with swallowtails and planar ends.
result Existence of maxfaces with infinitely many swallowtails and planar ends.
Solves constant mean curvature Dirichlet problem on catenoids with improved estimates.
problem Solving constant mean curvature Dirichlet problem on catenoidal necks.
method Found solutions in exponentially weighted Hölder spaces with non-integer weight.
result Improved estimate to γ=1 by comparing solutions with their limits on the disk.
We develop index theory on compact Riemannian spin manifolds with boundary in the case when the topological information is encoded by bundles which are supported away from the boundary. As a first application, we establish a "long neck principle" for a compact Riemannian spin -manifold with boundary , stating tha…
Constructs minimal immersions with singularities.
problem Minimal immersions with singularities in metric spaces.
method Constructs minimal immersions with catenoidal necks or floating disks converging to a singular point.
result Constructs minimal immersions with singularities.
We examine a Type-1 neck pinch singularity in simplicial Ricci flow (SRF) for an axisymmetric piecewise flat 3-dimensional geometry with 3-sphere topology. SRF was recently introduced as an unstructured mesh formulation of Hamilton's Ricci flow (RF). It describes the RF of a piecewise-flat simplicial geometry. In this …
3D Ricci flows have bounded diameter before Type I singularities.
problem Bounding the diameter of 3D Ricci flows before Type I singularities.
method Introduced a neck-region concept and proved packing measure Ahlfors regularity.
result Uniformly bounded diameter up to Type I singular time.
Researchers create minimal surfaces with Scherk ends and find catenoid limits.
problem Constructing minimal surfaces with specific end types and understanding their limits.
method Constructing families of embedded, singly periodic minimal surfaces with Scherk-type ends and analyzing their limits.
result The limit of the constructed surfaces are catenoid necks connecting planes, determined by Stieltjes polynomials.
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.
Study verifies Joyce's conjectures for circle-invariant Lagrangian surfaces.
problem Verifying Joyce's conjectures for specific Lagrangian surfaces.
method Continuation of Lagrangian mean curvature flow through finite time neck pinches.
result Flow converges to a chain of special Lagrangians, verifying conjectures.
The flow of symmetric spheres converges to a round sphere.
problem The behavior of symmetric hypersurfaces under inverse mean curvature flow.
method Localized parabolic maximum principle approach.
result The flow homothetically converges to a round sphere.
The paper constructs new non-trivial harmonic maps into higher-dimensional target manifolds.
problem Existence of non-trivial harmonic maps into higher-dimensional target manifolds.
method Perturbative argument, refined neck-analysis, energy identity, min-max problems.
result Construction of an infinite family of new null-homotopic -harmonic -spheres.
New proof classifies ancient flows in 3D space.
problem Classifying ancient noncollapsed flows in .
method Combining neck theorem and Harnack inequality rigidity.
result Directly establishes self-similarity of flows.
Paper uses 3-circle theorem to study Willmore surfaces and prove decay estimates.
problem Understanding Willmore surfaces and their properties.
method Applying the 3-circle theorem to analyze the second fundamental form of Willmore surfaces.
result Proves a decay estimate of the second fundamental form along the neck region.
Study elliptic operators on glued manifolds, reducing to finite-dimensional systems.
problem Mapping properties of elliptic operators in gluing problems.
method Reduction to finite-dimensional linear systems in the limit .
result Construction of Fredholm inverses with controlled norms.
Constructs surfaces with specific topologies and curvatures.
problem Creating surfaces with prescribed genus and ends.
method Using a family of constant mean curvature surfaces constructed in \cite{Kleene}, resolving tangency points over catenoidal necks.
result Complete embedded surfaces with freely prescribed genus and ends.
Real Lagrangian tori in are Hamiltonian isotopic to the Clifford torus.
problem Unknottedness of real Lagrangian tori in .
method Neck-stretching argument, Gromov's foliation theorem, Cieliebak-Schwingenheuer criterion.
result Real Lagrangian tori in are Hamiltonian isotopic to the Clifford torus.
Lectures on surface evolution through singularities.
problem Analyzing the mean curvature flow of surfaces and their singularities.
method Analysis of neck and conical singularities, using monotonicity formulas, epsilon-regularity, weak solutions, and blowup techniques.
result Unique evolution through neck singularities, nonuniqueness through conical singularities.
In earlier work, carrying out numerical simulations of the Ricci flow of families of rotationally symmetric geometries on , we have found strong support for the contention that (at least in the rotationally symmetric case) the Ricci flow for a ``critical'' initial geometry - one which is at the transition point bet…
Ancient pancakes solve mean curvature flow problem.
problem Mean curvature flow problem
method Constructing an embedded ancient solution as a stack of pancakes
result Embedded ancient solution to mean curvature flow