arXiv research

We give a classification of compact solitons for the pluriclosed flow on complex surfaces. First, by exploiting results from the Kodaira classification of surfaces, we show that the complex surface underlying a soliton must be Kähler except for the possibility of steady solitons on minimal Hopf surfaces. Then, we const…

Classifies solitons for surface diffusion flow of graphs.

Classifies solitons on invariant surfaces in solvable Lie group.

Study classifies minimal surfaces and solitons in hyperbolic 3-space as translation surfaces.

Characterizes ruled translating solitons in Minkowski 3-space.

Paper provides a formula for translating solitons and singular minimal surfaces.

The paper studies geometric properties of soliton surfaces using an extended Darboux frame field.

We study $3$-dimensional Ricci solitons which project via a semi-conformal mapping to a surface. We reformulate the equations in terms of parameters of the map; this enables us to give an ansatz for constructing solitons in terms of data on the surface. A complete description of the soliton structures on all the $3$-di…

Characterizes rotational solitons for curve shortening flow on revolution surfaces.

The Björling problem is explored for Born-Infeld solitons.

The paper examines soliton surfaces using a parallel transport frame field in 4D space.

Researchers create a family of solitons connecting a cigar to a sphere.

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

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

The paper classifies ruled surfaces in a specific space that move in a special way.

We show that the theory of isothermic surfaces in $\E^3$ -- one of the oldest branches of differential geometry -- can be reformulated within the modern theory of completely integrable (soliton) systems. This enables one to study the geometry of isothermic surfaces in $\E^3$ by means of powerful spectral methods availa…

This paper classifies all expanding Ricci solitons on surfaces.

Study identifies Kähler-Einstein, Kähler-Ricci soliton, and Sasaki-Einstein metrics on log del Pezzo surfaces.

The paper proves instability of translating λ-solitons and provides bounds on their length.

Using toric geometry we give an explicit construction of the compact steady solitons for pluriclosed flow first constructed in arXiv:1802.00170. This construction also reveals that these solitons are generalized Kähler in two distinct ways, with vanishing and nonvanishing Poisson structure. This gives the first example…

Study proves conditions for translating solitons to be planar.

The paper classifies solitons in the Heisenberg space.

The aim of this paper is to give a local description of affine surfaces, whose induced Blaschke structure is projectively flat. We show that such affine surfaces with constant Gauss affine curvature and indefinite induced Blaschke metric are described by soliton equations.

The article explores surfaces and soliton equations using spinors.

Study classifies bubbles of Type I singularities in Kähler-Ricci flow on compact surfaces.

Derives scalar reduction for generalized Kähler-Ricci solitons, proving uniqueness.

We show that a Born-Infeld soliton can be realised either as a spacelike minimal graph or timelike minimal graph over a timelike plane or a combination of both away from singular points. We also obtain some exact solutions of the Born-Infeld equation from already known solutions to the maximal surface equation. Further…

The paper classifies solitons in a curved product space.

Some aspects of the relation between differential geometry of curves and surfaces and multidimensional soliton equations is discussed. The connection between multidimensional soliton equations and Self-dual Yang-Mills equation is studied.

A connection between differential geometry and soliton equations is discussed

The height function of various surfaces decomposes into finite sums of scaled and translated versions of itself.

We give a survey of the following six closely related topics: (i) a general method for constructing a soliton hierarchy from a splitting of a loop algebra into positive and negative subalgebras, together with a sequence of commuting positive elements, (ii) a method---based on (i)---for constructing soliton hierarchies …

We show that the topological charge of the n-soliton solution of the sine-Gordon equation n is related to the genus g > 1 of a constant negative curvature compact surface described by this configuration. The relation is n=2(g-1), where n is even. The moduli space of complex dimension B(g)=3(g-1) corresponds precisely t…

Complete shrinking soliton found on a specific complex surface.

Symmetry groups help define solitons in curved spaces.

We show the properties of the blowup limits of \KRf solutions on Fano surfaces if Riemannian curvature is unbounded. As an application, on every toric Fano surface, we prove that \KRf converges to a Kähler Ricci soliton metric if the initial metric has toric symmetry. Therefore we give a new Ricci flow proof of existen…

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

In this paper we study the theory of self translating solitons of the mean curvature flow of immersed surfaces in the product space $\mathbb{H}^2\times\mathbb{R}$. We relate this theory to the one of manifolds with density, and exploit this relation by regarding these translating solitons as minimal surfaces in a confo…

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…

We give a review of the systematic construction of hierarchies of soliton flows and integrable elliptic equations associated to a complex semi-simple Lie algebra and finite order automorphisms. For example, the non-linear Schrödinger equation, the n-wave equation, and the sigma-model are soliton flows; and the equation…

The main objective of this thesis is the study of the evolution under the Ricci flow of surfaces with singularities of cone type. A second objective, emerged from the techniques we use, is the study of families of Ricci flow solitons in dimension 2 and 3. The Ricci flow is an evolution equation for Riemannian manifolds…

We establish a correspondence between Darboux's special isothermic surfaces of type (A,0,C,D) and the solutions of the second order PDE : uΔ(u)-|\nabla(u)|^{2}+Φ^{4}=s, s \in R. We then use the classical Darboux transformation for isothermic surfaces to construct a Bäcklund transformation for this equation and prove a …

Kähler soliton surfaces are typically toric under generic conditions.

The local structure of half conformally flat gradient Ricci almost solitons is investigated, showing that they are locally conformally flat in a neighborhood of any point where the gradient of the potential function is non-null. In opposition, if the gradient of the potential function is null, then the soliton is a ste…

The paper shows translating solitons in $\mathbb{R}^4$ have $SO(2)$ symmetry.

Solves Ricci flow on Riemann surfaces with measure initial data.

In this paper, we discuss a one parameter family of complex Born-Infeld solitons arising from a one parameter family of minimal surfaces. The process enables us to generate a new solution of the B-I equation from a given complex solution of a special type (which are abundant). We illustrate this with many examples. We …

The second del Pezzo surface is known by work of Tian-Zhu and Wang-Zhu to admit a unique Kaehler-Ricci soliton. Applying a method described in hep-th/0703057, we use Ricci flow to numerically compute that soliton metric. We numerically compute the value of its Perelman entropy (or Gaussian density).