arXiv research

Constructing translating solitons from Lagrangian Grim Reapers.

In this paper, we compute the first and second variation formulas for the F-functional of translating solitons and study the Hamiltonian L-stability of Lagrangian translating solitons. We prove that any Lagrangian translating soliton is Hamiltonian L-stable.

Proves better rigidity theorems for special solitons.

In this paper, we prove that any Lagrangian translating soliton is Lagrangian $L$-stable.

The paper examines translating solitons and their relation to Lagrangian mean curvature flows with zero Maslov class.

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…

New Lagrangian and special Lagrangian examples found in complex space.

The aim of this paper is to study variational properties for $f$-minimal Lagrangian submanifolds in Kähler manifolds with real holomorphy potentials. Examples of submanifolds of this kind incuding soliton solutions for Lagrangian mean curvature flow (LMCF). We derive second variation formula for $f$-minimal Lagrangians…

Using certain solutions of the curve shortening flow, including self-shrinking and self-expanding curves or spirals, we construct and characterize many new examples of translating solitons for mean curvature flow in complex Euclidean plane. They generalize the Joyce, Lee and Tsui ones \cite{JLT} in dimension two. The s…

The paper proves rigidity theorems for Type II singularities in Lagrangian flows.

We construct new examples of self-similar solutions and translating solitons for Lagrangian mean curvature flow by extending the method of Joyce, Lee and Tsui. Those examples include examples in which the Lagrangian angle is arbitrarily small as the examples of Joyce, Lee and Tsui.

In this paper, we construct finite blow-up examples for symplectic mean curvature flows and we study properties of symplectic translating solitons. We prove that, the Kähler angle $α$ of a symplectic translating soliton with $\max |A|=1$ satisfies that $\sup |α|>\fracπ{4}\frac{|T|}{|T|+1}$ where $T$ is the direction in…

In this paper, we investigate the parametric version and non-parametric version of rigidity theorem of spacelike translating solitons in pseudo-Euclidean space $\mathbb{R}^{m+n}_{n}$. Firstly, we classify $m$-dimensional complete spacelike translating solitons in $\mathbb{R}^{m+n}_{n}$ by affine technique and classical…

We prove an optimal control on the time-dependent measure of a measurable set under a reparametrized Lagrangian mean curvature flow of almost calibrated submanifolds in a Calabi-Yau manifold. Moreover we give a classification of those Lagrangian translating solitons in $\mathbb{C}^m$ that evolve by this reparametrized …

We consider the mean curvature flow of entire Lagrangian graphs with Lipschitz continuous initial data. Assuming only a certain bound on the Lipschitz norm of an initial entire Lagrangian graph in $\R^{2n}$, we show that the parabolic equation \eqref{PMA} for the Lagrangian potential has a longtime solution which is sm…

In this paper, we construct various examples of Lagrangian mean curvature flows in Calabi-Yau manifolds, using moment maps for actions of abelian Lie groups on them. The examples include Lagrangian self-shrinkers and translating solitons in the Euclidean spaces. Moreover, our method can be applied to construct examples…

Classifies regularity for Lagrangian mean curvature type equations.

Symplectic solitons rigid if bounded, study shows.

2-convex translating solitons are locally strictly convex.

Improved understanding of translating solitons using new techniques.

Characterizes ruled translating solitons in Minkowski 3-space.

Study proves conditions for translating solitons to be planar.

Classifies timelike translating solitons in Minkowski space.

Paper proves stability and Dirichlet problem for translating hypersurfaces.

The paper proves nonexistence results for translating solitons in r-mean curvature flow.

The study finds complete translating solitons for certain powers of Gaussian curvature in Riemannian products.

The paper explores new translating solitons and their relation to $Δ$-wings.

Study stabilizes translating solitons in hyperbolic space for MCF.

The paper shows translating solitons in $\mathbb{R}^4$ have $SO(2)$ symmetry.

We show that any complete $f$-stable translating soliton $M$ admits no codimension one cycle which does not disconnect $M$. As a corollary, it follows that any two dimensional complete $f$-stable translating soliton has genus zero.

We prove that any complete immersed two-sided mean convex translating soliton $Σ\subset \mathbb{R}^3$ for the mean curvature flow is convex. As a corollary it follows that an entire mean convex graphical translating soliton in $\mathbb{R}^3$ is the axisymmetric "bowl soliton". We also show that if the mean curvature of…

New curves defined by curvature powers studied for variational properties.

The paper proves convexity of certain solitons and expanders in high dimensions.

Paper provides a formula for translating solitons and singular minimal surfaces.

2D simply connected translating solitons in slabs are convex and have entropy < 3.

In this paper, we study entire spacelike translating solitons in Minkowski space. By constructing convex spacelike solutions to (1.3) in bounded convex domains, we obtain many entire smooth convex strictly spacelike translating solitons by prescribing boundary data at infinity.

In this paper, we prove a monotonicity formula and some Bernstein type results for translating solitons of hypersurfaces in $\re^{n+1}$, giving some conditions under which a trantranslating soliton is a hyperplane. We also show a gap theorem for the translating soliton of hypersurfaces in $R^{n+k}$, namely, if the $L^n…

The paper studies minimal surface flow and translating solitons, proving global solutions and convergence.

Paper finds convexity in translating solitons for concave flows.

Study classifies minimal surfaces and solitons in hyperbolic 3-space as translation surfaces.

The study classifies translating and self-expanding solitons in 3D space.

The paper classifies ruled surfaces in a specific space that move in a special way.

The paper classifies special solitons and shrinkers in Euclidean space.

Ancient solutions and translators identified for Lagrangian flow.

We study volume growth, entropy and stability for translating solitons of mean curvature flow. First, we prove that every complete properly immersed translator has at least linear volume growth. Then, by using Huisken's monotonicity formula, we compute the entropy of the grim reaper and the bowl solitons. We also give …

The paper classifies certain types of Lagrangian translators and self-expanders in complex 2-space.

The paper proves rigidity and vanishing theorems for translating solitons.

Study classifies translating solitons in Minkowski 3-space, revealing singularities.