arXiv research

Symmetry groups help define solitons in curved spaces.

The paper classifies special solitons and shrinkers in Euclidean space.

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…

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

The study examines constant weighted mean curvature hypersurfaces in shrinking Ricci solitons.

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

2-convex translating solitons are locally strictly convex.

Proves better rigidity theorems for special solitons.

New examples of solitons found using submersion techniques.

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

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

New proof shows symmetry for certain curved surfaces in higher dimensions.

The paper explores non-minimal solitons in the sphere with unique properties.

Study finds solitons on curved spaces with varying behavior.

Study translating solitons in Minkowski space with prescribed Gauss image.

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

Study of curvature blow-up in noncompact hypersurfaces using mean curvature flow.

Paper proves stability and Dirichlet problem for translating hypersurfaces.

The paper studies mean curvature flow in a Ricci flow background with extended Ricci flow.

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

In the present article we obtain classification results and topological obstructions for the existence of translating solitons of the mean curvature flow.

The paper classifies shapes of translating solitons for a specific flow.

Study stabilizes translating solitons in hyperbolic space for MCF.

Study applies Huisken formula to mean curvature flow in Ricci soliton background.

Study precise asymptotics of noncompact Type-IIb solutions to mean curvature flow.

In this paper, we study the existence, uniqueness and asymptotic behavior of rotationally symmetric translating solitons of the mean curvature flow in Minkowski space. We also study the asymptotic behavior and the strict convexity of general solitons of such flows.

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…

The paper classifies solitons for mean curvature flow in hyperbolic space.

The paper classifies solitons in a curved product space.

Mean curvature flow converges to a translating soliton with prescribed contact angle.

The aim of this paper is to introduce a notion of mean curvature flow soliton general enough to encompass target spaces of constant sectional curvature, Riemannian products or, in increasing generality, warped product spaces.

Researchers create nonconvex, non-soliton ancient flows in various dimensions.

The paper proves existence and classification of translating solitons in warped product manifolds.

Study complete gradient Ricci solitons with zero radial Weyl curvature.

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…

No breathers found for mean curvature flow in noncompact space.

Given a solution of the (backwards) Ricci flow one can construct a so called canonical soliton metric on space-time, introduced by E. Cabezas-Rivas and P. Topping. We observe that for a mean curvature flow within a (backwards) Ricci flow background, the space-time track of the mean curvature flow yields a canonical sol…

We study some properties of mean curvature flow solitons in general Riemannian manifolds and in warped products, with emphasis on constant curvature and Schwarzschild type spaces. We focus on splitting and rigidity results under various geometric conditions, ranging from the stability of the soliton to the fact that th…

This note surveys and compares results on the separation of variables construction for soliton solutions of curvature equations including the Kähler-Ricci flow and the Lagrangian mean curvature flow. In the last section, we propose some new generalizations in the Lagrangian mean curvature flow case.

We prove stability of rotationally symmetric translating solutions to mean curvature flow. For initial data that converge spatially at infinity to such a soliton, we obtain convergence for large times to that soliton without imposing any decay rates.

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…

We desingularise the union of $3$ Grim paraboloids along Costa-Hoffman-Meeks surfaces in order to obtain complete embedded translating solitons of the mean curvature flow with $3$ ends and arbitrary finite genus.

Huisken studied asymptotic behavior of a mean curvature flow in a Euclidean space when it develops a singularity of type I, and proved that its rescaled flow converges to a self-shrinker in the Euclidean space. In this paper, we generalize this result for a Ricci-mean curvature flow moving along a Ricci flow constructe…

In this paper we show that an immersed nontrivial translating soliton for mean curvature flow in $\mathbb{R}^{n+1}$($n=2,3)$ is a grim hyperplane if and only if it is mean convex and has weighted total extrinsic curvature of at most quadratic growth. For an embedded translating soliton $Σ$ with nonnegative scalar curva…

In this paper we obtain rigidity results and obstructions on the topology at infinity of translating solitons of the mean curvature flow in the Euclidean space. Our approach relies on the theory of f-minimal hypersurfaces.

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

Paper studies unique interior points and estimates for generalized translating soliton problems.

Following work of Ecker, we consider a weighted Gibbons-Hawking-York functional on a Riemannian manifold-with-boundary. We compute its variational properties and its time derivative under Perelman's modified Ricci flow. The answer has a boundary term which involves an extension of Hamilton's Harnack expression for the …