Study degenerations of Kähler-Einstein metrics on surfaces.
problem Understanding the geometry of Kähler-Einstein metrics on surfaces as they degenerate.
method Construct a Kähler-Einstein neck region to model degeneration.
result Provides a model for the limiting geometry of metrics in the family.
Compactness theory for biharmonic maps on degenerating Einstein manifolds.
problem Analyzing biharmonic maps on degenerating Einstein manifolds.
method Developed a compactness theory using asymptotic analysis over degenerating neck regions.
result Established a compactness theory for biharmonic maps with finitely many bubbles.
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.
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.
Stability of Morse index for harmonic maps on degenerating surfaces analyzed.
problem Analyzing stability of Morse index for harmonic maps on degenerating Riemann surfaces.
method Analysis of second variation of energy, identification of conditions for upper semicontinuity, explicit contribution of geodesics.
result Sharper control of spectrum of Jacobi operator, explicit contribution of geodesic segments to Morse index.
Proves energy quantization for surfaces with bounded index.
problem Energy quantization for Willmore surfaces with bounded index.
method Translated the question to the conformal Gauss map's perspective and showed convergence in specific regions.
result Conformal Gauss map converges to a light-like geodesic in De Sitter space in neck or collar regions.
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 non-degenerate harmonic functions on n-dimensional space.
problem Creating non-degenerate Z2-harmonic functions on Rn. method Using a variant of ellipsoidal coordinates, the construction is explicit and involves Lawlor's necks in Cn. result First known family of non-degenerate Z2-harmonic 1-forms with compact branching sets. 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.
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…
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.
Study of degenerating maps to Riemannian manifolds, proving asymptotic limits and existence of minimal cylinders.
problem Blow-up analysis and behavior of maps from degenerating surfaces to compact manifolds.
method Blow-up analysis, generalized energy identities, neck asymptotic limits, evolution system for minimal cylinders.
result Asymptotic limits of necks are geodesics or geodesic-like curves, confirming conjectures.
In this paper, we examine the dependence of standard gluing process for pseudoholomorphic curves under the change of the length T of the neck-region with respect to the cylindrical metrics associated to the given analytic coordinates near the punctures in the setting of bordered open Riemann surface with boundary pun…
We prove a bubble-neck decomposition together with an energy quantization result for sequences of Willmore surfaces into an arbitrary euclidian space with uniformly bounded energy and non-degenerating conformal type. We deduce the strong compactness of Willmore closed surfaces of a given genus modulo the Möbius group a…
Study compact Willmore surfaces without complex structure convergence, computing energy loss and geodesic lengths.
problem Compactness of Willmore surfaces without complex structure convergence.
method Compute energy loss in neck and geodesic lengths in Grassmannian G(2,n). result Limit of Gauss map image is a geodesic in G(2,n) with computable length. 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.
A theorem of Anderson and Bando-Kasue-Nakajima from 1989 states that to compactify the set of normalized Einstein metrics with a lower bound on the volume and an upper bound on the diameter in the Gromov-Hausdorff sense, one has to add singular spaces called Einstein orbifolds, and the singularities form as blow-downs …
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 n-harmonic n-spheres. Given a smooth polarized Riemann surface (X, L) endowed with a hyperbolic metric ω with cusp singularities along a divisor D, we show the L^2 projective embedding of (X, D) defined by L^k is asymptotically almost balanced in a weighted sense. The proof depends on sufficiently precise understanding of the behavior of …
We characterize a certain neck-pinching degeneration of (marked) CP1- structures on a closed oriented surface S of genus at least two. Namely, we consider a path Ct of CP1-structures on S leaving every compact subset in the deformation space of (marked) CP1-structures on S, such that its holonomy converges …
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 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.
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.
In this paper, we study the convergence of Yang-Mills-Higgs fields defined on fiber bundles over Riemann surfaces where the fiber is a compact symplectic manifold and the conformal structure of the Riemann surface is allowed to vary. We show that away from the nodes, the YMH fields converges, up to gauge, to a smooth Y…
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.
Under mean curvature flow, a closed, embedded hypersurface M(t) becomes singular in finite time. For certain classes of mean-convex mean curvature flows, we show the continuity of the first singular time T and the limit set "M(T)", with respect to initial data. We employ an Angenent-like neck-pinching argument to…
Classifies and constructs translators for curvature flows.
problem Understanding translating solitons in curvature flows.
method Developed rotational theory, introduced signed-neck framework.
result Classified and constructed catenoidal-type translators.
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…
Regular shrinkers describe blow-up limits of a finite-time singularity of the motion by curvature of planar network of curves. This follows from Huisken's monotonicity formula. In this paper, we show that there is only one regular shrinker with 2 closed regions. This regular shrinker is the Cisgeminate eye. Moreover, w…
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.
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.
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 3 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 …
Study on harmonic maps from surfaces to homogeneous spaces, focusing on bubble formation and geometric constraints.
problem Understanding the behavior of harmonic maps from surfaces to homogeneous spaces, especially in the presence of bubbles.
method Refined asymptotic expansions and obstruction relations for sequences developing a single bubble, geometric constraints for weakly conformal maps.
result New geometric constraints on the tangent planes of the limit map and bubble, depending on the dimensionality.
We will give a weak energy identity for Sacks-Uhlenbeck approximation of harmonic maps and calculate the length of the necks.
Study of Calabi-Yau manifold degenerations near complex structure limits.
problem Understanding polarized degenerations of Calabi-Yau manifolds.
method Improvement of metric convergence results on generic regions.
result Metric convergence for collapsing Ricci-flat Kähler metrics on generic regions.
Proves Arnold-Thom conjecture for surfaces' arrival times.
problem Existence of limit tangents for gradient flow lines of surfaces.
method Gradient flow lines of mean curvature flows with neck or cylindrical singularities.
result Proves Arnold's conjecture for all mean convex mean curvature flows of surfaces.
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. Investigates metric degeneracies on symplectic leaves using a generalized gradient flow.
problem Degeneracies in metrics on symplectic leaves of Poisson manifolds.
method Introduces the generalized double bracket (GDB) vector field to generalize gradient dynamics.
result Identifies admissible regions where the double bracket metric remains non-degenerate on symplectic leaves, enabling GDB as a gradient flow.
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.
Noise causes learning plateaus in neural networks.
problem Plateau phenomena in online learning due to vanishing gradients.
method Analysis of stochastic gradient descent in multi-layer perceptrons.
result Noise induces synchronisation leading to strong plateaus.