arXiv research

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

The paper classifies groups that can be isometry groups of infinite-genus hyperbolic surfaces.

Continuous epimorphisms between certain mapping class groups are induced by homeomorphisms.

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.

Study classifies 2-manifolds with special homeomorphism groups.

A connected combinatorial 2-manifold is called degree-regular if each of its vertices have the same degree. A connected combinatorial 2-manifold is called weakly regular if it has a vertex-transitive automorphism group. Clearly, a weakly regular combinatorial 2-manifold is degree-regular and a degree-regular combinator…

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.

We introduce the notion of complex $G_2$ manifold $M_{\mathbb C}$, and complexification of a $G_2$ manifold $M\subset M_{\mathbb C}$. As an application we show the following: If $(Y,s)$ is a closed oriented $3$-manifold with a $Spin^{c}$ structure, and $(Y,s)\subset (M, \varphi)$ is an imbedding as an associative subma…

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.

The study provides homological characterizations for $Q$-manifolds and $l_2$-manifolds.

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 define deformations of $RP^2$-manifolds.

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…

Game theory applied to splitting surfaces of compact 2-manifolds.

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…

Conditions for Penrose-Ward transformation on specific manifolds.

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…

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…

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 homeomorphism groups of telescoping 2-manifolds showing strong distortion.

This paper examines limits of Riemannian 2-manifolds with bounded curvature.

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

The paper constructs complex hyperbolic 2-manifolds with one cusp.

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…

Formula derived for $G_2$-manifolds, showing moduli spaces are incomplete.

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 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.

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.