arXiv research

Study on mixed Killing vector fields on Cigar Ricci-Bourguignon solitons.

Study on p-biharmonic maps from gradient Ricci solitons, focusing on 2D cigar soliton.

The only known example of collapsed three-dimensional complete gradient steady Ricci solitons so far is the 3D cigar soliton $N^2\times \mathbb{R}$, the product of Hamilton's cigar soliton $N^2$ and the real line $\mathbb{R}$ with the product metric. R. Hamilton has conjectured that there should exist a family of colla…

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

Analyzes harmonic and biharmonic maps from gradient Ricci solitons.

In this paper, we study the question if there is an isometric immersion of the cigar soliton into $R^3$. We show that the answer is negative. Similar result in higher dimensions is also true for steady Bryant solitons.

The study classifies steady Ricci solitons based on geometric conditions.

In this paper, we shall use the Kähler geometry formulation to study the global behavior of the Ricci flow on $R^2$. The geometric feature of our Ricci flow is that it has finite width. Our aim is to determine the limiting metric (which corresponds an eternal Ricci flow) obtained by L.F.Wu. We can use the classificatio…

We produce new non-Kähler complete steady gradient Ricci solitons whose asymptotics combine those of the Bryant solitons and the Hamilton cigar. We also obtain a family of complete Ricci-flat metrics with asymptotically locally conical asymptotics. Finally, we obtain numerical evidence for complete steady soliton struc…

Compact Ricci solitons on surfaces have at most two cone points, and are known as Hamilton's footballs. In this note we completely describe the degenerations of these footballs as one or both of the cone angles approaches zero. In particular, we show that Hamilton's famous non-compact cigar soliton is the Gromov--Hausd…

Researchers prove isoperimetric inequality in specific steady Ricci solitons.

We characterize complete nonnegatively curved steady gradient soliton with curvature in L^1. We show that there are isometric to a product (R^2,g_{cigar}) times(R^{n-2}, eucl))/Gamma where Gamma is a Bieberbach group of rank n-2. We prove also a similar local splitting result under weaker curvature assumptions.

This is an exposition of aspects of the result of Daskalopoulos and Sesum that any 2-dimensional complete noncompact ancient solution to Ricci flow with bounded positive scalar curvature and finite width must be the cigar soliton.

In this paper we prove new classification results for nonnegatively curved gradient expanding and steady Ricci solitons in dimension three and above, under suitable integral assumptions on the scalar curvature of the underlying Riemannian manifold. In particular we show that the only complete expanding solitons with no…

We construct many self-similar and translating solitons for Lagrangian mean curvature flow, including self-expanders and translating solitons with arbitrarily small oscillation on the Lagrangian angle. Our translating solitons play the same role as cigar solitons in Ricci flow, and are important in studying the regular…

We study the non Ricci flat gradient steady Kähler Ricci soliton with non-negative Ricci curvature and weak integrability condition of the scalar curvature $S$, namely $\underline{\lim}_{r\to \infty} r^{-1}\int_{B_r} S=0$, and show that it is a quotient of $Σ\times \mathbb{C}^{n-1-k}\times N^k$, where $Σ$ and $N$ denot…

In this paper we consider a perturbation of the Ricci solitons equation proposed by J. P. Bourguignon in \cite{jpb1}. We show that these structures are more rigid then standard Ricci solitons. In particular, we prove that there is only one complete three--dimensional, positively curved, Riemannian manifold satisfying $…

There are described equations for a pair comprising a Riemannian metric and a Killing field on a surface that contain as special cases the Einstein Weyl equations (in the sense of D. Calderbank) and a real version of a special case of the Abelian vortex equations, and it is shown that the property that a metric solve t…

New Kähler solitons found that are not $U(n)$-invariant.

Let (M,g) be a Riemannian manifold with an isometric action of the Lie group G. Let g_G be a left invariant metric on G. Consider the diagonal G action on the product $M \times G$ with the metric g+g_G. In this paper we calculate the formula for the metric h on the quotient space $(M \times G) / G$; the map from g to h…

We study sequences of 3-dimensional solutions to the Ricci flow with almost nonnegative sectional curvatures and diameters tending to infinity. Such sequences may arise from the limits of dilations about singularities of Type IIb. In particular, we study the case when the sequence collapses, which may occur when dilati…

We show that the Cigar metric on $\mathbb{C}$ is an example of real analytic Kähler manifold with globally defined and positive Calabi's diastasis function which cannot be Kähler immersed into any (finite or infinite dimensional) complex space form.

Lectures detail field theory dynamics and exact WKB analysis.

We consider the initial value problem $u_t = Δ\log u$, $u(x,0) = u_0(x)\ge 0$ in $\R^2$, corresponding to the Ricci flow, namely conformal evolution of the metric $u (dx_1^2 + dx_2^2)$ by Ricci curvature. It is well known that the maximal (complete) solution $u$ vanishes identically after time $T= \frac 1{4π} \int_{\R^…

Type II (ancient) solutions to the Ricci flow on surfaces are not yet classified. It is conjectured that the Rosenau solution and the cigar are the only solutions, modulo scaling. In this paper, we mainly study the backward limit and the circumference at spatial infinity of Type II ancient solutions on noncompact surfa…

The Ricci flow has been of fundamental importance in mathematics, most famously though its use as a tool for proving the Poincaré Conjecture and Thurston's Geometrization Conjecture. It has a parallel life in physics, arising as the first order approximation of the Renormalization Group flow for the nonlinear sigma mod…

The paper classifies a type of solitons in Euclidean spaces.

Study on shrinking solitons of generalized Ricci flow.

New examples of solitons found as warped products.

Study on Yamabe solitons with applications and structure elucidation.

We classify Algebraic Ricci Solitons of three-dimensional Lorentzian Lie groups. All algebraic Ricci solitons that we obtain are sol-solitons. In particular, we prove that, contrary to the Riemannian case, Lorentzian Ricci solitons need not to be algebraic Ricci solitons. We classify Algebraic Ricci Solitons of three-d…

The study examines Riemann solitons and almost solitons on specific Kenmotsu manifolds.

Study on $η$-Ricci-Yamabe solitons on Riemannian submersions.

Study on geometric properties of second Ricci solitons.

Study on Ricci-like solitons and gradient solitons on specific manifolds.

The study characterizes spacetimes with specific solitons in $f(\mathcal{R})$-gravity.

We prove some results for the solitons of the Ricci-Bourguignon flow, generalizing corresponding results for Ricci solitons. Taking motivation from Ricci almost solitons, we then introduce the notion of Ricci-Bourguignon $almost$ solitons and prove some results about them which generalize previous results for Ricci alm…

Extends soliton theory to non-compact cases.

Paper shows constant σk-curvature for quasi k-Yamabe solitons.

3D solitons classified into specific types.

Researchers study solitons on homogeneous manifolds, proving properties of specific types of solitons.

The paper characterizes Ricci solitons on the Poincaré upper half plane.

Characterizes $*$-$k$-Ricci-Yamabe solitons on Kenmotsu manifolds.

The paper explores almost Ricci solitons on Finsler spaces, proving conditions for their existence.

New families of Ricci solitons found with collapsing volume.

The study classifies h-almost Ricci-Yamabe solitons in various paracontact manifolds.

Study on warped product Yamabe solitons with constant fiber curvature.

Study on para-Sasakian metrics and their solitons.