arXiv research

New expanding Ricci solitons found starting in dimension four.

No expanding breathers found in noncompact Ricci flows with certain curvature conditions.

Study on the spectrum of drift Laplacian on Ricci expanders.

Study classifies 4D Ricci solitons with specific curvature conditions.

Sharp curvature estimates for expanding Ricci solitons in various dimensions.

Unique continuation result for expanding Ricci solitons.

Paper proves rigidity for Ricci solitons with specific conditions.

We first investigate the asymptotics of conical expanding gradient Ricci solitons by proving sharp decay rates to the asymptotic cone both in the generic and the asymptotically Ricci flat case. We then establish a compactness theorem concerning nonnegatively curved expanding gradient Ricci solitons.

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

New example of non-Kähler soliton with Kähler-like behavior at infinity.

We establish Carleman inequalities for the weighted laplacian associated to an expanding gradient Ricci soliton. As a consequence, a unique continuation at infinity is proved for asymptotically Ricci flat Ricci expanders. The obstruction at infinity is a symmetric 2-tensor defined on the link of the corresponding asymp…

In this paper, we prove that expanding gradient Ricci solitons with (positively) pinched Ricci curvature are trivial ones. Namely, they are either compact or flat.

The paper pinches curvature in expanding Ricci solitons.

Constructs expanding gradient Ricci solitons with unique properties.

This paper classifies all expanding Ricci solitons on surfaces.

We investigate the possibility of desingularizing a positively curved metric cone by an expanding gradient Ricci soliton with positive curvature operator. This amounts to study the deformation of such geometric structures. As a consequence, we prove that the moduli space of conical positively curved gradient Ricci expa…

Study shows expanding Ricci solitons from specific metric cones.

Study cohomogeneity one expanding Ricci solitons on specific topologies.

In this paper, we study gradient Ricci expanding solitons $(X,g)$ satisfying $Rc=cg+D^2f,$ where $Rc$ is the Ricci curvature, $c<0$ is a constant, and $D^2f$ is the Hessian of the potential function $f$ on $X$. We show that for a gradient expanding soliton $(X,g)$ with non-negative Ricci curvature, the scalar curva…

New nontrivial breathers found for Ricci flow on noncompact manifolds.

We prove the weak stability of expanding gradient Ricci solitons with positive curvature operator and quadratic curvature decay at infinity.

We compute the second variation of the Ricci expander entropy and briefly discuss the linear stability of compact negative Einstein manifolds.

Unique steady and expanding solitons with spherical links identified.

We show that an expanding gradient Ricci solitons which is asymptotic to a cone at infinity in a certain sense must be rotationally symmetric.

We introduce the notion of Canonical Expanding Ricci Soliton, and use it to derive new Harnack inequalities for Ricci flow. This viewpoint also gives geometric insight into the existing Harnack inequalities of Hamilton and Brendle.

We produce new examples of non-Kähler complete expanding gradient Ricci solitons on trivial vector bundles over a product of Einstein manifolds with positive scalar curvature.

The paper shows how to stabilize perturbed Kähler-Ricci solitons.

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…

We show that at the level of formal expansions, any compact Riemannian manifold is the sphere at infinity of an asymptotically conical gradient expanding Ricci soliton.

Study expanding gradient Ricci solitons with Euclidean base.

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.

New findings on gradient expanding Ricci solitons with finite scalar curvature ratio.

The paper establishes curvature estimates for solitons in higher dimensions.

In this paper, we study how to get the Ricci expanders from W+-functional through the heat kernel estimate of the conjugate heat equation to the type III singularity of Ricci flow. The Gaussian upper and lower bounds are established for the related heat kernel in accordance to the interesting work of Cao-Zhang for the …

The paper estimates curvature for specific 4D Ricci solitons.

We show that expanding Kähler-Ricci solitons which have positive holomorphic bisectional curvature and are asymptotic to Kähler cones at infinity must be the U(n)-rotationally symmetric expanding solitons constructed by Cao.

In this paper, the author discusses the eigenvalues and entropies under the harmonic-Ricci flow, which is the Ricci flow coupled with the harmonic map flow. We give an alternative proof of results for compact steady and expanding harmonic-Ricci breathers. In the second part, we derive some monotonicity formulas for eig…

Classifies ancient and expanding Ricci flows with specific groups.

The Kähler-Ricci flow near conical singularities is described with a $C/t$ curvature bound.

Study of special Lorentzian Lie groups with 4D isometry group, finding all are non-gradient expanding Ricci solitons.

We study the geometry at infinity of expanding gradient Ricci solitons of dimension greater than two with finite asymptotic curvature ratio without curvature sign assumptions. We mainly prove that they have a cone structure at infinity.

We show that gradient shrinking, expanding or steady Ricci solitons have potentials leading to suitable reference probability measures on the manifold. For shrinking solitons, as well as expanding soltions with nonnegative Ricci curvature, these reference measures satisfy sharp logarithmic Sobolev inequalities with low…

In this paper we discuss Perelman's Lambda-functional, Perelman's Ricci shrinker entropy as well as the Ricci expander entropy on a class of manifolds with isolated conical singularities. On such manifolds, a singular Ricci de Turck flow preserving the isolated conical singularities exists by our previous work. We prov…

The study provides estimates for curvature of gradient Ricci solitons.

Study expanding Ricci solitons on vector bundles, reducing to Higgs bundle equations.

This paper classifies Ricci solitons in complex hyperbolic spaces.

We produce new non-Kähler, non-Einstein, complete expanding gradient Ricci solitons with conical asymptotics and underlying manifold of the form $\R^2 \times M_2 \times \cdots \times M_r$, where $r \geq 2$ and $M_i$ are arbitrary closed Einstein spaces with positive scalar curvature. We also find numerical evidence for…

We give necessary and sufficient conditions for a Kähler equivariant resolution of a Kähler cone, with the resolution satisfying one of a number of auxiliary conditions, to admit a unique asymptotically conical (AC) expanding gradient Kähler-Ricci soliton. In particular, it follows that for any $n\in\mathbb{N}_{0}$ and…