arXiv research

Study finds solutions for degenerate affine curve shortening flow.

Symmetry groups help define solitons in curved spaces.

The two main theorems of this paper provide a characterization of hyperbolic affine iterated function systems defined on Rm. Atsushi Kameyama (Distances on Topological Self-Similar Sets, Proceedings of Symposia in Pure Mathematics, Volume 72.1, 2004) asked the following fundamental question: given a topological self-si…

Two self-similar solutions found for time-like hypersurfaces in Minkowski spacetime.

The paper lists all self-similar solutions for a flow in 2D space.

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

The paper examines self-similar solutions in warped products.

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.

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

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

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

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…

The paper examines the stability of two spherical self-similar solutions in Minkowski spacetime.

We propose a construction which transforms a self-similar zipper in $\mathbb R^n$ to a self-affine zipper $\mathbb R^{n+1}$ whose attractor is a smooth curve.

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.

Smooth knots can be embedded into a specific Menger continuum.

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.

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

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 $\R^{n+1}$.

New self-similarity for Einstein vacuum equations identified.

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…

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…

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 …

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.

Study on mean curvature flow in a cone, proving existence and homogenization.

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…

New proof of self-similar solutions for inverse mean curvature flow.

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…

Study on transverse Ricci solitons on compact foliated manifolds.

Study of naked singularities in Einstein vacuum equations using new self-similarity.

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.

The paper examines properties of self-affine Sierpiński sponges using metric invariants.

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…

We consider the class of self-similar Gaussian stochastic volatility models, and compute the small-time (near-maturity) asymptotics for the corresponding asset price density, the call and put pricing functions, and the implied volatilities. Unlike the well-known model-free behavior for extreme-strike asymptotics, small…