arXiv research

Study on shrinking solitons of generalized Ricci flow.

Paper classifies 3D breathers and generalizes Ricci soliton results.

Symmetries in shrinking Ricci solitons spread outward.

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

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…

New proof of shrinking gradient Ricci soliton rigidity.

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

The paper proves various inequalities on gradient shrinking Ricci solitons.

The paper proves manifold isometries for certain gradient Ricci solitons.

5D shrinking solitons with bounded curvature are rigid.

Complete shrinking soliton found on a specific complex surface.

Sharp Gaussian bounds derived for Schrödinger kernel on Ricci 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…

Paper proves properties of minimal hypersurfaces in specific solitons.

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

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 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…

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

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

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 rigidity for Ricci solitons with specific conditions.

Study on 4D Ricci solitons with specific curvature properties.

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

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.

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.

Study shows Sasaki solitons with harmonic Weyl tensor are spheres.

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.

Simply-connected shrinking Kähler-Ricci solitons are proven.

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.

Gradient shrinking solitons from Ricci flows terminating in cones.

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

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 $\…

Paper proves a Liouville theorem for solitons with constant curvature.

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…

Sharp upper diameter limit found for Ricci solitons.

4D gradient solitons with constant curvature are rigid.

New gradient Ricci solitons found for $SU(2)$ invariants.

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 …

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

Haslhofer and Müller proved a compactness Theorem for four-dimensional shrinking gradient Ricci solitons, with the only assumption being that the entropy is uniformly bounded from below. However, the limit in their result could possibly be an orbifold Ricci shrinker. In this paper we prove a compactness theorem for non…

Study on 4D solitons with curvature constraints.

Estimates heat equation on shrinking Ricci solitons with uniform bounds.

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…

The paper examines the geometry and topology of Sasaki-Ricci solitons, proving they are either connected at infinity or compact.

In this paper we derive a precise estimate on the growth of potential functions of complete noncompact shrinking solitons. Based on this, we prove that a complete noncompact gradient shrinking Ricci soliton has at most Euclidean volume growth. The latter result can be viewed as an analog of the well-known theorem of Bi…

As of today, there are very few known complete shrinking Ricci solitons in dimension 4, and all examples discovered so far are Kähler and/or Einstein. In this note, we prove that any four dimensional J-invariant gradient shrinking Ricci solitons satisfy a differential form identity relating Kählerity annd Einstein-ness…

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…