arXiv research

In this paper we study potential function of gradient steady Ricci solitons. We prove that infimum of potential function decays linearly; in particular, potential function of rectifiable gradient steady Ricci solitons decays linearly. As a consequence, we show that a gradient steady Ricci soliton with bounded potential…

New classification of gradient steady Ricci solitons with vanishing D-tensor.

3D steady gradient Ricci solitons are all O(2)-symmetric.

Study 4D steady gradient Ricci solitons reducing to 3D manifolds.

Study classifies 4D Ricci solitons with specific curvature conditions.

Paper proves volume growth estimate for steady gradient Ricci solitons.

Gradient steady Ricci solitons are natural generalizations of Ricci-flat manifolds. In this article, we prove a curvature gap theorem for gradient steady Ricci solitons with nonconstant potential functions; and a curvature gap theorem for Ricci-flat manifolds, removing the volume growth assumptions in known results.

In this paper we prove that any $n$-dimensional ($n\ge 4$) complete Bach-flat gradient steady Ricci soliton with positive Ricci curvature is isometric to the Bryant soliton. We also show that a three-dimensional gradient steady Ricci soliton with divergence-free Bach tensor is either flat or isometric to the Bryant sol…

New decay estimates for scalar curvature of steady gradient Ricci solitons.

Study steady gradient Ricci solitons with cylindrical tangent flows at infinity.

Survey paper analyzes curvature estimates for 4D gradient Ricci solitons.

Unique soliton found on resolved cones.

The paper examines gradient Ricci solitons on orbifolds and proves their rigidity properties.

The study examines 4D steady gradient Ricci solitons with nonnegative curvature away from a compact set.

Study on curvature decay in steady Ricci solitons, proving dichotomy.

Study proves rigidity of certain gradient steady Ricci solitons with harmonic Weyl curvature.

The paper studies steady solitons with curvature decay and proves their smoothness.

In this paper, we classify n-dimensional (n>2) complete noncompact locally conformally flat gradient steady solitons. In particular, we prove that a complete noncompact non-flat conformally flat gradient steady Ricci soliton is, up to scaling, the Bryant soliton.

In this paper, we derive certain curvature estimates for 4-dimensional gradient steady Ricci solitons either with positive Ricci curvature or with scalar curvature decay.

New steady gradient Ricci solitons found with specific symmetry.

Paper reviews and proves volume growth estimates for different types of gradient Ricci solitons.

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

Study on Kähler-Ricci solitons on toric manifolds, proving rigidity.

In this paper, we give a description for steady Ricci solitons with a linear decay of sectional curvature. In particular, we classify all 3-dimensional steady Ricci solitons and 4-dimensional $κ$-noncollpased steady Ricci solitons with nonnegative sectional curvature under the linear curvature decay.

Paper studies fundamental groups of certain Ricci solitons.

We study the stability of non compact steady and expanding gradient Ricci solitons. We first show that strict linear stability implies dynamical stability. Then we give various sufficient geometric conditions ensuring the strict linear stability of such gradient Ricci solitons.

In this very short note we prove a lower bound for the scalar curvature of certain steady gradient Ricci solitons.

Let $(M,g,f)$ be a 3-dimensional complete steady gradient Ricci soliton. Assume that $M$ is rectifiable, that is, the potential function can be written as $f=f(r)$, where $r$ is a distance function. Then, we prove that $M$ is isometric to (1) a quotient of $\mathbb{R}^3$, or (2) the Bryant soliton. In particular, we sh…

New steady gradient Ricci solitons found on specific four-manifolds.

We show that for an $n$ dimensional complete non Ricci flat gradient steady Ricci soliton with potential function $f$ bounded above by a constant and curvature tensor $Rm$ satisfying $\overline{\lim}_{r\to \infty} r|Rm|<\frac{1}{5}$, then $|Rm|\leq Ce^{-r}$ for some constant $C>0$, improving a result of [36]. For any f…

We show that the linear trace Harnack quadratic on a steady gradient Ricci soliton satisfies the heat equation. Similar result holds for shrinkers. We also present an interpolation between Perelman's and Cao--Hamilton's Harnacks on a steady soliton.

In this paper, we give a delay estimate of scalar curvature for a complete non-compact expanding (or steady) gradient Ricci soliton with nonnegative Ricci curvature. As an application, we prove that any complete non-compact expanding (or steady) gradient Kähler-Ricci solitons with positively pinched Ricci curvature sho…

New explicit Calabi-Yau metrics and Kähler-Ricci solitons found on complex n-space.

We describe the structure of the Ricci tensor on a locally homogeneous Lorentzian gradient Ricci soliton. In the non-steady case, we show the soliton is rigid in dimensions three and four. In the steady case, we give a complete classification in dimension three.

In this paper we study volume growth of gradient steady Ricci solitons. We show that if the potential function satisfies a uniform condition, then the soliton has at most Euclidean volume growth.

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

The study classifies steady Ricci solitons based on geometric conditions.

We show for a complete noncompact steady Ricci soliton that there exists a sequence {x_i} of points tending to infinity such that |Rc|(x_i) limits to zero.

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

In this paper, we prove that any $κ$-noncollapsed gradient steady Ricci soliton with nonnegative curvature operator and horizontally $ε$-pinched Ricci curvature must be rotationally symmetric. As an application, we show that any $κ$-noncollapsed gradient steady Ricci soliton $(M^n, g,f)$ with nonnegative curvature oper…

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…

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

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…

Let (M,g) be a steady gradient Ricci soliton of dimension n \geq 4 which has positive sectional curvature and is asymptotically cylindrical. Under these assumptions, we show that (M,g) is rotationally symmetric. In particular, our result applies to steady gradient Ricci solitons in dimension 4 which are κ-noncollapsed …

New steady Kähler-Ricci solitons found on orbifolds.

The main purpose of this paper is to show that a normalized non-steady gradient Ricci soliton (M,g,f,λ) of dimension n is trivial if and only if its scalar curvature S satisfies the equality S=λ(f+(n/2)).

In this paper, we show that steady or shrinking complete gradient Yamabe solitons with finite total scalar curvature and non-positive Ricci curvature are Ricci flat. Moreover, under certain pinching condition for Ricci curvature, we show that steady or shrinking complete gradient Yamabe solitons with finite total scala…

Researchers discover a new family of 3D solitons that are flying wings.