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.

Paper proves volume growth estimate for steady gradient Ricci solitons.

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

Unique soliton found on resolved cones.

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…

Study classifies 4D Ricci solitons with specific curvature conditions.

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

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.

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

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

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

The paper classifies 3D complete gradient Yamabe solitons.

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

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

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

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

New steady gradient Ricci solitons found with specific symmetry.

Paper shows triviality criterion for certain Ricci solitons.

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

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…

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.

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.

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

The study classifies specific types of solitons with bounded scalar curvature.

Paper studies fundamental groups of certain Ricci solitons.

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.

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.

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

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

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

Unique steady and expanding solitons with spherical links identified.

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.

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.

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…

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

The study classifies steady Ricci solitons based on geometric conditions.

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

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

Characterizes gradient Yamabe solitons with specific conditions.

Researchers found cylindrical steady gradient solitons in 3D.

New steady Kähler-Ricci solitons found on orbifolds.

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.

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

This research shows that steady solitons in higher dimensions always reduce at infinity.

Constructs new steady gradient Ricci solitons for higher dimensions.