arXiv research

Study on shrinking solitons of generalized Ricci flow.

In this paper, we classify n-dimensional (n>3) complete Bach-flat gradient shrinking Ricci solitons. More precisely, we prove that any 4-dimensional Bach-flat gradient shrinking Ricci soliton is either Einstein, or locally conformally flat hence a finite quotient of the Gaussian shrinking soliton $R^4$ or the round cyl…

Symmetries in shrinking Ricci solitons spread outward.

Paper classifies 3D breathers and generalizes Ricci soliton results.

The study proves rotationally symmetric property of certain shrinking gradient Yamabe solitons.

Complete shrinking soliton found on a specific complex surface.

5D shrinking solitons with bounded curvature are rigid.

New proof of shrinking gradient Ricci soliton rigidity.

The study examines four-dimensional gradient Ricci solitons and their properties.

In this paper, we will give a local version of the Hamilton-Ivey type pinching estimate of the gradient shrinking soliton with vanishing Weyl tensor, and then give a complete classification on gradient shrinking solitons with vanishing Weyl tensor.

The study shows ends of shrinking gradient $ρ$-Einstein solitons are non-parabolic.

The paper proves various inequalities on gradient shrinking Ricci solitons.

In this paper we classify the four dimensional gradient shrinking solitons under certain curvature conditions satisfied by all solitons arising from finite time singularities of Ricci flow on compact four manifolds with positive isotropic curvature. As a corollary we generalize a result of Perelman on three dimensional…

Sharp Gaussian bounds derived for Schrödinger kernel on Ricci solitons.

We first show that a Kähler cone appears as the tangent cone of a complete expanding gradient Kähler-Ricci soliton with quadratic curvature decay with derivatives if and only if it has a smooth canonical model (on which the soliton lives). This allows us to classify two-dimensional complete expanding gradient Kähler-Ri…

The study proves triviality and rigidity results for Ricci solitons and estimates their conjugate radius.

We prove that there does not exist non-constant positive $f$-harmonic function on the complete gradient shrinking Ricci solitons. We also prove the $L^{p}(p\geq 1 \ {\rm or}\ 0<p\leq 1)$ Liouville theorems on the complete gradient shrinking Ricci solitons.

Paper proves properties of minimal hypersurfaces in specific solitons.

We give two new proofs of Perelman's theorem that shrinking breathers of Ricci flow on closed manifolds are gradient Ricci solitons, using the fact that the singularity models of type I solutions are shrinking gradient Ricci solitons and the fact that non-collapsed type I ancient solutions have rescaled limits being sh…

Undergraduate thesis explores topological barriers to compact Ricci solitons in 4D.

Study shows Sasaki solitons with harmonic Weyl tensor are spheres.

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

Uniqueness proven for specific types of geometric structures.

Unique shrinking gradient Kähler-Ricci solitons found on non-compact toric manifolds.

In this paper, we first apply an integral identity on Ricci solitons to prove that closed locally conformally flat gradient Ricci solitons are of constant sectional curvature. We then generalize this integral identity to complete noncompact gradient shrinking Ricci solitons, under the conditions that the Ricci curvatur…

We prove that a gradient shrinking Ricci soliton with fourth order divergence-free Riemannian tensor is rigid. For the $4$-dimensional case, we show that any gradient shrinking Ricci soliton with fourth order divergence-free Riemannian tensor is either Einstein, or a finite quotient of the Gaussian shrinking soliton $\…

We prove rigidity theorems for shrinking gradient Ricci solitons supporting the Heisenberg-Pauli-Weyl uncertainty principle with the sharp constant in $\mathbb{R}^n$. In addtion, we partially give analogous rigidity results of the Caffarelli-Kohn-Nirenberg inequalities on shrinking Ricci solitons.

4D gradient solitons with constant curvature are rigid.

In this paper we prove that any asymptotically cylindrical gradient shrinking Ricci soliton is isometric to a cylinder.

In this paper, we prove some rigidity theorems for shrinking gradient Ricci solitons with nonnegative sectional curvature.

5D shrinking Ricci solitons with constant scalar curvature are rigid.

Alternative proof for 4D shrinking Ricci solitons with constant scalar curvature.

In this paper, we prove that any complete shrinking gradient Kähler-Ricci solitons with positive orthogonal bisectional curvature must be compact. We also obtain a classification of the complete shrinking gradient Kähler-Ricci solitons with nonnegative orthogonal bisectional curvature.

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…

We show that the only complete shrinking gradient Ricci solitons with vanishing Weyl tensor are quotients of the standard ones. This gives a new proof of the Hamilton-Ivey-Perel'man classification of 3-dimensional shrinking gradient solitons. We also prove a classification for expanding gradient Ricci solitons with con…

Study on 4D Ricci solitons with specific curvature properties.

Paper proves rigidity for Ricci solitons with specific conditions.

Paper proves a Liouville theorem for solitons with constant curvature.

In this paper, we have proved that if a complete conformally flat gradient shrinking Ricci soliton has linear volume growth or the scalar curvature is finitely integrable and also the reciprocal of the potential function is subharmonic, then the manifold is isometric to the Euclidean sphere. As a consequence, we have s…

We prove a splitting theorem for complete gradient Ricci soliton with nonnegative curvature and establish a rigidity theorem for codimension one complete shrinking gradient Ricci soliton in $\mathbb R^{n+1}$ with nonnegative Ricci curvature.

The main purpose of this article is to provide an alternate proof to a result of Perelman on gradient shrinking solitons. In dimension three we also generalize the result by removing the $κ$-non-collapsing assumption. In high dimension this new method allows us to prove a classification result on gradient shrinking sol…

We discuss some geometric conditions under which a complete noncompact shrinking gradient Ricci soliton will split at infinity.

Gradient shrinking solitons from Ricci flows terminating in cones.

In the first part of the paper we derive integral curvature estimates for complete gradient shrinking Ricci solitons. Our results and the recent work of Lopez-Rio imply rigidity of gradient shrinking Ricci solitons with harmonic Weyl tensor. In the second part of the paper we address the issue of existence of harmonic …

Simply-connected shrinking Kähler-Ricci solitons are proven.

Sharp upper diameter limit found for Ricci solitons.

The gradient shrinking $ρ$-Einstein soliton is a triple $(M^n,g,f)$ such that $R_{ij}+f_{ij}=(ρR+λ) g_{ij},$ where $(M^n,g)$ is a Riemannian manifold, $λ>0, ρ\in\mathbb{R}\setminus\{0\}$ and $f$ is the potential function on $M^n$. In this paper, using algebraic curvature estimates and the Yamabe-Sobolev inequality, w…

The study finds a lower bound for the diameter of gradient ρ-Einstein solitons.