arXiv research

We prove that the only self-similar surfaces of Euclidean 3-space which are foliated by circles are the self-similar surfaces of revolution discovered by S. Angenent and that the only ruled, self-similar surfaces are the cylinders over planar self-similar curves.

Lipschitz equivalence of self-similar sets is an important area in the study of fractal geometry. It is known that two dust-like self-similar sets with the same contraction ratios are always Lipschitz equivalent. However, when self-similar sets have touching structures the problem of Lipschitz equivalence becomes much …

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

Skeleton is a new notion designed for constructing space-filling curves of self-similar sets. It is shown in [Dai, Rao and Zhang, Space-filling curves of self-similar sets (II): Edge-to-trail substitution rule,https://doi.org/10.1088/1361-6544/ab1275] that for a connected self-similar set, space-filling curves can be c…

Study properties of self-similar continua with finite intersection property.

The article contains a construction of a self-similar dendryte which cannot be the attractor of any self-similar zipper.

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

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

Wave maps with noise can lead to self-similar blowup from arbitrary initial data.

Study on self-similar surfaces and their mapping class groups generated by involutions.

Study finds solutions to flows by negative curvature powers.

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

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.

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 …

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.

In [9] Kaimanovich introduced the concept of augmented tree on the symbolic space of a self-similar set. It is hyperbolic in the sense of Gromov, and it was shown in [13] that under the open set condition, a self-similar set can be identified with the hyperbolic boundary of the tree. In the paper, we investigate in det…

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…

In this paper, we study two classes of planar self-similar fractals $T_\varepsilon$ with a shifting parameter $\varepsilon$. The first one is a class of self-similar tiles by shifting $x$-coordinates of some digits. We give a detailed discussion on the disk-likeness ({\it i.e., the property of being a topological disk}…

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…

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…

We present new examples of complete embedded self-similar surfaces under mean curvature by gluing a sphere and a plane. These surfaces have finite genus and are the first examples of self-shrinkers in $\mathbb R^3$ that are not rotationally symmetric. The strategy for the construction is to start with a family of initi…

We consider the heat flow of corotational harmonic maps from $\mathbb R^3$ to the three-sphere and prove the nonlinear asymptotic stability of a particular self-similar shrinker that is not known in closed form. Our method provides a novel, systematic, robust, and constructive approach to the stability analysis of self…

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

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.

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…

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

The study explores homeomorphism groups of self-similar 2-manifolds, including the 2-sphere and Cantor set.

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…

We prove that an irreducible lattice in a semisimple algebraic group is virtually isomorphic to an arithmetic lattice if and only if it admits a faithful self-similar action on a rooted tree of finite valency.

A novel approach to analyzing time series generated by complex systems, such as markets, is presented. The basic idea of the approach is the {\it Law of Self-Similar Evolution}, according to which any complex system develops self-similarly. There always exist some internal laws governing the evolution of a system, say …

We propose a new approach for analyzing price fluctuations in their strongly correlated regime ranging from minutes to months. This is done by employing a self-similarity assumption for the magnitude of coarse-grained price fluctuation or volatility. The existence of a Cramer function, the characteristic function for s…

Wavelet scattering spectra model non-Gaussian time-series, proving scale invariance for self-similar processes.

New self-similarity for Einstein vacuum equations identified.

We construct helicoid-like embedded minimal disks with axes along self-similar curves modeled on logarithmic spirals. The surfaces have a self-similarity inherited from the curves and the nature of the construction. Moreover, inside of a "logarithmic cone", the surfaces are embedded.

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…

In this paper, we introduce the notion of asymptotic self-similar sets on general doubling metric spaces by extending the notion of self-similar sets, and determine their Hausdorff dimensions, which gives an extension of Balogh and Rohner 's result. This is carried out by introducing the notions of almost similarity ma…

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.

Robust method estimates self-similarity for mammogram images, improving cancer detection.

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.

We introduce the continuum self-similar tree (CSST) and characterize it topologically. We apply this to answer a question of Curien about the topology of the continuum random tree (CRT). We also give a topological characterization of other trees with branch points of finite or infinite valences.

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.

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…

Fractal Lipschitz-Killing curvature measures C^f_k(F,.), k = 0, ..., d, are determined for a large class of self-similar sets F in R^d. They arise as weak limits of the appropriately rescaled classical Lipschitz-Killing curvature measures C_k(F_r,.) from geometric measure theory of parallel sets F_r for small distances…