arXiv research

The paper extends energy identities and neck existence for ε-harmonic maps.

A multi-neck spacetime wormhole is constructed with a simple metric tensor.

The paper proves the existence of maxfaces with multiple swallowtails and planar ends.

Lipid necks, large curvature bridges, are shown to be metastable.

Paper proposes a smart neck-band for detecting neck postures using integrated kinematic and kinetic data.

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 with energy bounds and neck domains.

Study neck pinches in Lagrangian flows, proving stability and introducing new singularities.

Paper solves long neck problem on odd-dimensional spin manifolds.

Paper proves energy identity and no-neck property for special harmonic maps.

Derives 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.

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.

The paper constructs solutions with infinite-time singularities in Lagrangian mean curvature flow.

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.

Classifies 85 tie knots into mathematical categories.

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.

Mean curvature flow shows singularities on smooth surfaces.

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.

Proves principles and estimates for initial data sets in Einstein equations.

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 constructs new non-trivial harmonic maps into higher-dimensional target manifolds.

Constructs minimal immersions with singularities.

We prove that on any symplectic manifold whose symplectic form represents a rational cohomology class there exists a sequence of compatible almost complex structures whose Nijenhuis energy (the $L^2$-norm of the Nijenhuis tensor) tends to zero. The sequence is obtained by stretching the neck around a Donaldson hypersur…

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 …

Study on singularities in Lagrangian mean curvature flow with special Lagrangian cones.

3D Ricci flows have bounded diameter before Type I singularities.

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…

Researchers create minimal surfaces with Scherk ends and find catenoid limits.

Constructing translating solitons from Lagrangian Grim Reapers.

We study the Ricci flow on $\mathbb{R}^{4}$ starting at an SU(2)-cohomogeneity 1 metric $g_{0}$ whose restriction to any hypersphere is a Berger metric. We prove that if $g_{0}$ has no necks and is bounded by a cylinder, then the solution develops a global Type-II singularity and converges to the Bryant soliton when su…

Study verifies Joyce's conjectures for circle-invariant Lagrangian surfaces.

No compact surfaces with specific curvature can exist near singular limits.

CAST models time-varying treatment effects in cancer patients.

The paper constructs harmonic 1-forms on 3-manifolds with cylindrical necks.

New proof classifies ancient flows in 3D space.

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…

In this paper, we prove the mean-convex neighborhood conjecture for neck singularities of the mean curvature flow in $\mathbb{R}^{n+1}$ for all $n\geq 3$: we show that if a mean curvature flow $\{M_t\}$ in $\mathbb{R}^{n+1}$ has an $S^{n-1}\times \mathbb{R}$ singularity at $(x_0,t_0)$, then there exists an $\varepsilon…

Paper uses 3-circle theorem to study Willmore surfaces and prove decay estimates.

Constructs surfaces with specific topologies and curvatures.

Real Lagrangian tori in $S^2 imes S^2$ are Hamiltonian isotopic to the Clifford torus.

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…

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…