Study proves neck properties for smooth maps with bounded energy.
problem Understanding neck formation in smooth maps with bounded energy.
method Proved energy identity and no neck property for extrinsic polyharmonic maps.
result Established neck properties for maps with bounded total energy.
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 ε-harmonic case. result Energy identity and no-neck property established for ε- and α-harmonic maps. Study investigates singularity formation in α-Yang-Mills-Higgs fields on spheres.
problem Singularity formation in α-Yang-Mills-Higgs fields on spheres. method Established α-energy identity, no-neck property through Hodge decomposition and new conservation law. result Unified and quantitative framework for singularity formation in variational gauge theories.
Study analyzes convergence of harmonic maps into compact locally CAT(1) spaces.
problem Analyzing convergence of harmonic maps with energy bounds.
method Bubble tree convergence, exploiting local convexity properties of CAT(1) spaces.
result Energy quantization and no-neck property demonstrated for harmonic maps.
The study examines the behavior of maps near the boundary of surfaces.
problem Analyzing the behavior of approximate harmonic maps near the boundary of surfaces.
method Blow-up analysis and energy identity for maps from surfaces to manifolds with free boundary.
result Energy identity and no neck property hold during the blow-up process.
Harmonic map flow's singularity properties proven with Lojasiewicz inequalities.
problem Finite-time singularities of harmonic map flow in critical dimensions.
method Proving a weighted Lojasiewicz inequality.
result Continuity of body map and no-neck property for bubble-tree decompositions.
The paper discusses mean curvature flow with surgeries for 2-convex hypersurfaces, focusing on neck detection and gluing.
problem Mean curvature flow with surgeries for 2-convex hypersurfaces.
method Establishing neck detection, gluing cross sections, and using harmonic spherical parametrisation.
result Uniqueness, existence, and overlapping properties for normal parametrisations on (ε,k)-cylindrical hypersurface necks. 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.
Paper analyzes harmonic maps blow-up neck regions.
problem Understanding blow-up regions in harmonic maps.
method Proved refined estimates in neck regions.
result New estimate reveals neck shape and inequality about nullity and index.
Study Ricci flow on R^4 starting at a specific metric, finding singularities and minimal spheres.
problem Analyzing Ricci flow on R4 starting from a specific metric. method Ricci flow on R4, focusing on metrics with no necks and bounded by a cylinder. result The flow develops a global Type-II singularity and converges to the Bryant soliton.
No compact surfaces with specific curvature can exist near singular limits.
problem Existence of surfaces with prescribed mean curvature near singular limits.
method Analyzing mappings and Delaunay tori in Euclidean 3-space.
result No parametric surface with the specified curvature exists near singular limits.
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 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.
We show that asymptotically Schwarzschildean 3-manifolds cannot contain minimal surfaces obtained by perturbative deformations of a Euclidean catenoid, no matter how small the ADM mass of the ambient space and how large the neck of the catenoid itself. Such an obstruction is sharply three-dimensional and ceases to hold…
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.
No finite-time singularities in Yang-Mills flow in 4D.
problem Finite-time singularities in Yang-Mills flow.
method Weighted energy identity and sharp decay estimates.
result Long-time existence of Yang-Mills flow in 4D.
We explore geometric aspects of bubble convergence for harmonic maps. More precisely, we show that the formation of bubbles is characterised by the local excess of curvature on the target manifold. We give a universal estimate for curvature concentration masses at each bubble point and show that there is no curvature l…
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.
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.
We exhibit many examples of closed symplectic manifolds on which there is an autonomous Hamiltonian whose associated flow has no nonconstant periodic orbits (the only previous explicit example in the literature was the torus T^2n (n\geq 2) with an irrational symplectic structure). The underlying smooth manifolds of our…
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…
The paper shows mean curvature flow keeps diameter bounded under certain conditions.
problem Proving the bounded diameter of hypersurfaces under mean curvature flow.
method Use of Lojasiewicz inequalities and solution of mean-convex neighbourhood conjecture.
result The intrinsic diameter stays uniformly bounded as the flow approaches the first singular time.
We find calibrated submanifolds in neck manifolds. Particularly, we obtain a calibrated submanifold in the Lagrangian self-expander constructed by Joyce, Lee and Tsui.
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.
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.
Harmonic map flow preserves almost-holomorphic maps without singularities.
problem Preserving almost-holomorphic maps without singularities.
method Harmonic map flow applied to almost-holomorphic maps.
result No singularities or necks appear in the limit at singular times.
The paper studies a geometric model combining Kaluza-Klein, Yang-Mills, and Dirac actions.
problem Analyzing the geometric and analytic aspects of a complex model.
method Investigates geometric and analytic properties of a model combining Kaluza-Klein, Yang-Mills, and Dirac actions.
result For a sequence of approximate solutions on surfaces with uniformly bounded energies, energy identities and the no-neck property hold.
The paper examines Yang-Mills-Higgs pairs on vector bundles and proves stability and energy identity.
problem Stability and energy identity of Yang-Mills-Higgs pairs on vector bundles.
method Bubble-neck decomposition and analysis of weakly stable pairs.
result A sequence of Yang-Mills-Higgs pairs converges to a Yang-Mills-Higgs pair with uniformly bounded energy.
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 L1-bounds on Riemann curvature tensor. EasiCS improves neck function assessment by objectively classifying cervical spondylosis.
problem Subjective and coarse-grained neck function assessment methods.
method Developed clustering algorithms on sEMG data to objectively classify cervical spondylosis.
result EasiCS outperforms existing seven algorithms overall.
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.
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.
The study proves all limit flows are self-similar under specific conditions.
problem The selfsimilarity of limit flows in mean curvature flow.
method Proof of selfsimilarity conditions for mean convex surfaces and neck singularities.
result All limit flows are self-similar if and only if there are finitely many spherical singularities.
This paper detects necks in two-convex hypersurfaces under a specific flow.
problem Detecting necks in two-convex hypersurfaces under a new flow.
method Adjusting gradient estimates and following Huisken-Sinestrari's surgery algorithm.
result Classifies two-convex surfaces undergoing the G-flow.
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.
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.
Proves convergence of gradient Ricci shrinkers with uniform bounds.
problem Compactness and energy concentration in gradient Ricci shrinkers.
method Bubble-tree convergence and local energy analysis.
result No energy concentrates in neck regions, leading to a local diffeomorphism finiteness theorem.
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…
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 …
The paper studies neck-pinching of CP1-structures on surfaces, describing their limits.
problem Characterizing the degeneration of CP1-structures on surfaces. method Analyzing a path of CP1-structures leaving every compact subset, converging holonomy in the PSL(2, C)-character variety. result The limit of the path Ct is described in terms of developing maps, holomorphic quadratic differentials, and pleated surfaces. The paper proves a conjecture about mean curvature flows in higher dimensions.
problem Proving a conjecture about mean curvature flows in higher dimensions.
method Classifying ancient Brakke flows and introducing a novel moving plane method.
result Proves the mean-convex neighborhood conjecture for higher dimensional mean curvature flows.
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.