arXiv research

Symmetry groups help define solitons in curved spaces.

Ancient solutions to curve shortening flow are constructed and analyzed.

In this paper we use the anholonomic frames method to construct exact solutions for vacuum 5D gravity with metrics having off-diagonal components. The solutions are in general anisotropic and possess interesting features such as an anisotropic warp factor with respect to the extra dimension, or a gravitational scaling/…

Study on transverse Ricci solitons on compact foliated manifolds.

In this article we give a brief survey of breather and soliton solutions to the Ricci flow and prove a no breather and soliton theorem for homogeneous solutions.

The paper classifies invariant gradient $k$-Yamabe solitons in pseudo-Euclidean spaces.

The Ward equation, also called the modified 2+1 chiral model, is obtained by a dimension reduction and a gauge fixing from the self-dual Yang-Mills field equation on $R^{2,2}$. It has a Lax pair and is an integrable system. Ward constructed solitons whose extended solutions have distinct simple poles. He also used a li…

Classifies solitons for surface diffusion flow of graphs.

It is well-known that the LIE(Locally Induction Equation) admit soliton-type solutions and same soliton solutions arise from different and apparently irrelevant physical models. By comparing the solitons of LIE and Killing magnetic geodesics, we observe that these solitons are essentially decided by two families of iso…

New gravitational solitons and infinite topological manifolds found.

Study of 4D Ricci solitons with symmetry, finding precise geometric asymptotics.

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

The study finds solitons for curve shortening flow on hyperbolic plane.

The paper studies minimal surface flow and translating solitons, proving global solutions and convergence.

Paper examines properties of specific solutions to Yamabe flow.

New curvature condition proves rigidity of Bryant Ricci solitons.

We study the evolution of homogeneous Ricci solitons under the bracket flow, a dynamical system on the space of all homogeneous spaces of dimension n with a q-dimensional isotropy, which is equivalent to the Ricci flow for homogeneous manifolds. We prove that algebraic solitons (i.e. the Ricci operator is a multiple of…

Ancient solutions to Ricci flow in higher dimensions are mostly cylinders or solitons.

Two ancient solutions to Gauss curvature flow are identified for cylinders.

We give two new proofs of Perelman's theorem that shrinking breathers of Ricci flow on closed manifolds are gradient Ricci solitons, using the fact that the singularity models of type I solutions are shrinking gradient Ricci solitons and the fact that non-collapsed type I ancient solutions have rescaled limits being sh…

Constructs explicit solutions to Spin(7)-structures gradient flow.

We investigate Riemannian (non-Kahler) Ricci flow solutions that develop finite-time Type-I singularities and present evidence in favor of a conjecture that parabolic rescalings at the singularities converge to singularity models that are shrinking Kahler-Ricci solitons. Specifically, the singularity model for these so…

Study on how soliton equations form singularities using L,A,B-triples.

Study ancient solutions on noncompact steady Ricci solitons, proving types of ancient solutions.

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.

New families of Ricci solitons found with collapsing volume.

Study 4D solitons with specific curvature properties, proving curvature bounds and classifying solutions.

In this paper, we study entire spacelike translating solitons in Minkowski space. By constructing convex spacelike solutions to (1.3) in bounded convex domains, we obtain many entire smooth convex strictly spacelike translating solitons by prescribing boundary data at infinity.

In this paper we investigate the behavior of three-dimensional homogeneous solutions of the cross curvature flow using Riemannian groupoids. The Riemannian groupoid technique, introduced by John Lott, allows us to investigate the long term behavior of collapsing solutions of the flow, producing soliton solutions in the…

Ricci solitons are natural generalizations of Einstein metrics. They are also special solutions to Hamilton's Ricci flow and play important roles in the singularity study of the Ricci flow. In this paper, we survey some of the recent progress on Ricci solitons.

Study of solitons in Laplacian flow on 7-manifolds.

Ozawa solution describes surface deformation from Davey-Stewartson II equation.

The paper classifies periodic solitons in curve flows on the light-cone.

In the previous article (J. Geom. Phys. {\bf 43} (2002) 146), we show the hyperelliptic solutions of a loop soliton as a study of a quantized elastica. This article gives some functional relations in a loop soliton as a quantized elastica.

We discuss the geometry of homogeneous Ricci solitons. After showing the nonexistence of compact homogeneous and noncompact steady homogeneous solitons, we concentrate on the study of left invariant Ricci solitons. We show that, in the unimodular case, the Ricci soliton equation does not admit solutions in the set of l…

Using the `Riemann Problem with zeros' method, Ward has constructed exact solutions to a (2+1)-dimensional integrable Chiral Model, which exhibit solitons with nontrivial scattering. We give a correspondence between what we conjecture to be all pure soliton solutions and certain holomorphic vector bundles on a compact …

Conditions for a soliton's dual form to be harmonic or Ricci harmonic are derived.

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…

Study Born-Infeld solitons and solve Björling problem for them.

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

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 consider a class of time dependent finite energy multi-soliton solutions of the U(N) integrable chiral model in $(2+1)$ dimensions. The corresponding extended solutions of the associated linear problem have a pole with arbitrary multiplicity in the complex plane of the spectral parameter. Restrictions of these exten…

Study properties of hyperbolic Yamabe solitons in submanifolds.

We provide the classification of locally conformally flat gradient Yamabe solitons with positive sectional curvature. We first show that locally conformally flat gradient Yamabe solitons with positive sectional curvature have to be rotationally symmetric and then give the classification and asymptotic behavior of all r…

We consider Ricci flow of complete Riemannian manifolds which have bounded non-negative curvature operator, non-zero asymptotic volume ratio and no boundary. We prove scale invariant estimates for these solutions. Using these estimates, we show that there is a limit solution, obtained by scaling down this solution at a…

New findings on $κ$-solutions with round cylinder as asymptotic shrinker.

Study on scalar curvature decay in four-dimensional steady solitons.

We consider the Laplacian "co-flow" of $G_2$-structures: $\frac{d}{dt} ψ= - Δ_d ψ$ where $ψ$ is the dual 4-form of a $G_2$-structure $φ$ and $Δ_d$ is the Hodge Laplacian on forms. This flow preserves the condition of the $G_2$-structure being coclosed ($dψ=0$). We study this flow for two explicit examples of coclosed $…