Study confirms neck condition for budding vesicles of any shape.
problem Understanding the shape of budding vesicles with two-phase domains.
method Derived shape equation and linking conditions, refined conjecture, proved for asymmetric vesicles.
result Mean curvature condition holds for all membrane segments adjacent to the neck in budding vesicles.
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.
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.
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.
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.
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.
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.
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.
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 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.
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. 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.
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. The paper shows continuity of singular times and limit sets for certain mean-convex flows.
problem Understanding the behavior of singularities in mean curvature flow.
method Employing an Angenent-like neck-pinching argument and using Andrews' α-non-collapsed condition and Colding-Minicozzi uniqueness of tangent flows.
result Shows continuity of first singular time and limit set with respect to initial data.
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.
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.
We find calibrated submanifolds in neck manifolds. Particularly, we obtain a calibrated submanifold in the Lagrangian self-expander constructed by Joyce, Lee and Tsui.
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.
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.
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.
The paper proves a 'long neck principle' for Riemannian spin manifolds with positive scalar curvature.
problem Establishing a 'long neck principle' for Riemannian spin manifolds with boundary.
method Developed index theory on compact Riemannian spin manifolds with boundary and applied it to prove the 'long neck principle'.
result The distance between the support of the differential of a strictly area decreasing map and the boundary of a manifold is bounded.
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 …
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…
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. CAST models time-varying treatment effects in cancer patients.
problem Estimating treatment effects at fixed time points limits understanding of dynamic changes over time.
method CAST combines parametric and non-parametric methods to model continuous time-varying treatment effects.
result CAST reveals how treatment effects rise, peak, and decline over the follow-up period.
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.
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.
Flow preserves Lagrangian condition in Calabi-Yau manifolds.
problem Preserving Lagrangian condition in Calabi-Yau manifolds with boundary.
method Introduced mixed Dirichlet-Neumann boundary condition for Lagrangian mean curvature flow.
result Proved preservation of Lagrangian condition under flow.
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.
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.
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.
New proof classifies ancient flows in 3D space.
problem Classifying ancient noncollapsed flows in R3. method Combining neck theorem and Harnack inequality rigidity.
result Directly establishes self-similarity of flows.
This review reports some key results in theoretical investigations on configurations of lipid membranes and presents several challenges in this field which involve (i) exact solutions to the shape equation of lipid vesicles; (ii) exact solutions to the governing equations of open lipid membranes; (iii) neck condition o…
New curvature condition preserves Ricci flow in higher dimensions, proving flow extension beyond singularities.
problem Proving Ricci flow extension in higher dimensions with curvature control.
method New curvature condition, neck-like curvature pinching estimate, surgery procedure.
result Proves flow extension beyond singularities in higher dimensions.
Deep learning improves radiotherapy by automating head and neck cancer organ segmentation.
problem Manual delineation of radio-sensitive organs in head and neck cancer radiotherapy is time-consuming and variably performed.
method A 3D U-Net deep learning architecture trained on clinical CT scans and expert segmentations.
result Deep learning model achieves expert-level performance in delineating 21 distinct OARs.
Paper examines exponential decay and smoothness of pseudoholomorphic curves moduli space.
problem Dependence of gluing process on neck-region length in bordered open Riemann surfaces.
method Establishes exponential decay of T-derivatives of glued solutions under neck-region length change. result Smoothness of Kuranishi structure on moduli space of pseudoholomorphic curves.
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.
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.
In earlier work, carrying out numerical simulations of the Ricci flow of families of rotationally symmetric geometries on S3, 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…