arXiv research

The paper analyzes self-similar solutions for mean curvature flow in 3D.

Study finds solutions to flows by negative curvature powers.

Paper proves rigidity for self-similar solutions in 3D flows.

Self-similar solutions to geometric flows are stable under small perturbations.

In this paper, we consider affine self-similar solutions for the affine curve shortening flow in the Euclidean plane. We obtain the equations of all affine self-similar solutions up to affine transformations and solve the equations or give descriptions of the solutions for the degenerate case. Some new special solution…

In this letter, two explicit self-similar solutions to a graph representation of time-like extremal hypersurfaces in Minkowski spacetime $\mathbb{R}^{1+3}$ are given. Meanwhile, there is an untable eigenvalue in the linearized time-like extremal hypersurfaces equation around two explicit self-similar solutions.

Classifies self-similar curve shortening flows in hyperbolic 2-space.

We give a classification of all self-similar solutions to the curve shortening flow in the plane.

Classifies self-similar solutions for heat equations with positive speed.

The study proves uniqueness and symmetry of self-similar solutions in warped product spaces.

This article gives an alternative approach to the self-shrinking and self-expanding solutions of the curve shortening flow, which are related to singularity formation of the mean curvature flow. The motivation for the self-similar solutions arises from natural area preserving rescaling. Further we describe the self-sim…

In this paper, we obtain a complete list of all self-similar solutions of inverse mean curvature flow in $\mathbb{R}^2$.

We classify all Hamiltonian stationary Lagrangian surfaces in complex Euclidean plane which are self-similar solutions of the mean curvature flow.

Study on self-similar solutions of supercritical Fujita equation, proving entropy and energy gap.

Curvature flow and inverse curvature flow solutions on 2D light cone identified.

We introduce the mean curvature flow of curves in the Minkowski plane $\mathbf R^{1,1}$ and give a classification of all the self-similar solutions. In addition, we describe five other exact solutions to the flow.

In this paper we study self-similar solutions in warped products satisfying $F-\mathcal{F}=\bar{g}(λ(r)\partial_{r},ν)$, where $\mathcal{F}$ is a nonnegative constant and $F$ is in a class of general curvature functions including powers of mean curvature and Gauss curvature. We show that slices are the only closed stri…

By the curve shortening flow, the only closed embedded contracting self-similar solutions are circles: we give a very short and intuitive geometric proof of this basic and classical result using an idea of Gage.

We carry out the first main step towards the construction of new examples of complete embedded self-similar surfaces under mean curvature flow. An approximate solution is obtained by taking two known examples of self-similar surfaces and desingularizing the intersection circle using an appropriately modified singly per…

The paper constructs and analyzes self-similar blowup solutions for a wave map equation.

We classify the self-similar solutions to a class of Weingarten curvature flow of connected compact convex hypersurfaces, isometrically immersed into space forms with non-positive curvature, and obtain a new characterization of a sphere in a Euclidean space $\R^{n+1}$.

Paper classifies ruled surfaces in Lorentz-Minkowski space for a specific flow.

New self-similarity for Einstein vacuum equations identified.

The self-similar solutions to the mean curvature flows have been defined and studied on the Euclidean space. In this paper we initiate a general treatment of the self-similar solutions to the mean curvature flows on Riemannian cone manifolds. As a typical result we extend the well-known result of Huisken about the asym…

In this survey article, we discuss some topics on self-similar solutions to the Ricci flow and the mean curvature flow. Self-similar solutions to the Ricci flow are known as Ricci solitons. In the first part of this paper we discuss a lower diameter bound for compact manifolds with shrinking Ricci solitons. Such a boun…

We study the contraction of a convex immersed plane curve with speed (1/α)k^{α}, where αin(0,1] is a constant and show that, if the blow-up rate of the curvature is of type one, it will converge to a homothetic self-similar solution. We also discuss a special symmetric case of type two blow-up and show that it converge…

We develop a local theory for the construction of singular spacetimes in all spacetime dimensions which become asymptotically self-similar as the singularity is approached. The techniques developed also allow us to construct and classify exact self-similar solutions which correspond to the formal asymptotic expansions …

This paper is devoted to the study of the singularity phenomenon of timelike extremal hypersurfaces in Minkowski spacetime $\mathbb{R}^{1+3}$. We find that there are two explicit lightlike self-similar solutions to a graph representation of timelike extremal hypersurfaces in Minkowski spacetime $\mathbb{R}^{1+3}$, the …

Classifies solitons for surface diffusion flow of graphs.

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.

Estimates the rate of convergence of mean curvature flow solutions.

We confirm a well-known conjecture that the round sphere is the only compact, embedded self-similar shrinking solution to the mean curvature flow with genus $0$. More generally, we show that the only properly embedded self-similar shrinkers in $\mathbb{R}^3$ with vanishing intersection form are the sphere, the cylinder…

We study the Dirichlet problem associated to the equation for self-similar surfaces for graphs over the Euclidean plane with a disk removed. We show the existence of a solution provided the boundary conditions on the boundary circle are small enough and satisfy some symmetries. This is the second step towards the const…

Study on transverse Ricci solitons on compact foliated manifolds.

This work addresses the {\em singularity formation} of complete non-compact solutions to the conformally flat Yamabe flow whose conformal factors have {\em cylindrical behavior at infinity}. Their singularity profiles happen to be {\em Yamabe solitons}, which are {\em self-similar solutions} to the fast diffusion equat…

The paper defines Hesse solitons and explores their properties on Hessian manifolds.

In this paper, employing a new inequality, we show that under certain curvature pinching condition, the strictly convex closed smooth self-similar solution of $σ_k^α$-flow must be a round sphere. We also obtain a similar result for the solutions of $F=-\langle X, e_{n+1}\rangle \, (*)$ with a non-homogeneous function $…

The study finds that only round spheres shrink self-similarly under certain curvature flows.

We prove the existence of a (spectrally) stable self-similar blow-up solution $f_0$ to the heat flow for corotational harmonic maps from $\mathbb R^3$ to the three-sphere. In particular, our result verifies the spectral gap conjecture stated by one of the authors and lays the groundwork for the proof of the nonlinear s…

I classify spacelike self-similar shrinking solutions of the mean curvature flow in pseudo-euclidean space in arbitrary codimension, if the mean curvature vector is not a null vector and the principal normal vector is parallel in the normal bundle. Moreover, I exclude the existence of such self-shrinkers in several cas…

We prove the existence of self-similar expanding solutions of the curvature flow on planar networks where the initial configuration is any number of half-lines meeting at the origin. This generalizes recent work by Schnürer and Schulze which treats the case of three half-lines. There are multiple solutions, and these a…

Study classifies ruled surfaces from mean curvature flow solutions.

We show the uniqueness of strictly convex closed smooth self-similar solutions to the $α$-Gauss curvature flow with $(1/n) < α< 1+(1/n)$. We introduce a Pogorelov type computation, and then we apply the strong maximum principle. Our work combined with earlier works on the Gauss Curvature flow imply that the $α$-Gauss c…

Study Ricci flows on manifolds, proving they behave like self-similar solutions and confirming a conjecture.

We consider a mean curvature flow in a cone, that is, a hypersurface in a cone which moves toward the opening with normal velocity equaling to the mean curvature, and the contact angle between the hypersurface and the cone boundary being $\varepsilon$-periodic in its position. First, by constructing a family of self-si…

New proof shows origin-centred balls are unique solutions to curvature problems.

Study on subgroups' evolution in 3D Lie groups using mean curvature flow.

We construct new ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as $t \to -\infty$, to two self-similar complete non-compact solutions to the Yamabe flow moving in opposite directions. They are type I ancient solutions.