arXiv research

Constructing translating solitons from Lagrangian Grim Reapers.

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

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

Butscher, D. Lee, Y. Lee, and Joyce constructed a special Lagrangian submanifold by gluing a Lawlor neck into a transverse intersection point of two special Lagrangian submanifolds. We prove a uniqueness theorem for the gluing of flat special Lagrangian tori of real dimension 3 in a flat complex torus of complex dimens…

Explains examples of Lagrangian flow with circle symmetry.

Flow preserves Lagrangian condition in Calabi-Yau manifolds.

We study almost-calibrated, $O(n)$-equivariant Lagrangian mean curvature flow in $\mathbb{C}^n$, and prove structural theorems about the Type I and Type II blowups of finite-time singularities. In particular, we prove that any Type I blowup of such a flow must be a special Lagrangian pair of transversely intersecting p…

We prove two main results: (a) Suppose $L$ is a closed, embedded, exact special Lagrangian $m$-fold in ${\mathbb C}^m$ for $m\ge 3$ asymptotic at infinity to the union $Π_1\cupΠ_2$ of two transverse special Lagrangian planes $Π_1,Π_2$ in ${\mathbb C}^m$. Then $L$ is one of the explicit 'Lawlor neck' family of examples …

Researchers create non-degenerate harmonic functions on n-dimensional space.

We give a necessary and sufficient condition for a special Lagrangian submanifold in C^n constructed by Lawlor being also special Lagrangian in C^n with the Fubini-Study form.

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

We prove two gluing theorems for special Lagrangian (SL) conifolds in complex space C^m. Conifolds are a key ingredient in the compactification problem for moduli spaces of compact SLs in Calabi-Yau manifolds. In particular, our theorems yield the first examples of smooth SL conifolds with 3 or more planar ends and the…

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

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

We find calibrated submanifolds in neck manifolds. Particularly, we obtain a calibrated submanifold in the Lagrangian self-expander constructed by Joyce, Lee and Tsui.

Study shows how neck pinches occur in Lagrangian flows and their continuation.

In this paper we prove the existence of families of n-dimensional complete embedded minimal submanifolds of C^n with a prescribed configuration of k>1 asymptotic planes. These submanifolds are obtained by desingularizing the intersection of the asymptotes, using a gluing theorem applied to a generalization of a special…

We construct many self-similar and translating solitons for Lagrangian mean curvature flow, including self-expanders and translating solitons with arbitrarily small oscillation on the Lagrangian angle. Our translating solitons play the same role as cigar solitons in Ricci flow, and are important in studying the regular…

A second order family of special Lagrangian submanifolds of complex m-space is a family characterized by the satisfaction of a set of pointwise conditions on the second fundamental form. For example, the set of ruled special Lagrangian submanifolds of complex 3-space is characterized by a single algebraic equation on t…

This is the second in a series of papers constructing explicit examples of special Lagrangian submanifolds in C^m. The first paper was math.DG/0008021, which studied special Lagrangian m-folds with large symmetry groups. The third is math.DG/0010036, which uses ideas from this paper to construct families of special Lag…

Proves minimality of tensor varieties, generalizing previous results.

Paper detects duality obstruction in smooth calibrations.

New construction of Fukaya-Seidel categories using complex gradient flow equation.

Suppose $M_{1}$ and $M_{2}$ are two special Lagrangian submanifolds of $\Rtn$ with boundary that intersect transversally at one point $p$. The set $M_{1} \cup M_{2}$ is a singular special Lagrangian variety with an isolated singularity at the point of intersection. Suppose further that the tangent planes at the interse…

Nearby pinwheels are isotopic, solving Arnold's conjecture.

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

Study on harmonic maps from surfaces with energy bounds and neck domains.

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

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

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

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

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 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.

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

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 …

Classifies 85 tie knots into mathematical categories.

We show that every area-minimizing hypercone and every oriented Lawlor cone in [Law91] can be realized as a tangent cone at a point of some homologically area-minimizing singular compact submanifold. In particular this generalizes the result of N. Smale [Sma99].

This paper proves area-minimizing cones over Grassmannian manifolds.

We will give a weak energy identity for Sacks-Uhlenbeck approximation of harmonic maps and calculate the length of the necks.

We show the area-minimality property of all homogeneous area-minimizing hypercones in Euclidean spaces (classified by Lawlor) following Lawson's original idea in his 72' Trans. A.M.S. paper "The equivariant Plateau problem and interior regularity". Moreover, each of them enjoys (coflat) calibrations singular only at th…

Study controls curvature in Ricci flows using necks.

Mean curvature flow shows singularities on smooth surfaces.

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

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