arXiv research

New G2-structures found on Lie groups with strong structural conditions.

We explicitly describe the solution of the G$_2$-Laplacian flow starting from an extremally Ricci-pinched closed G$_2$-structure on a compact 7-manifold and we investigate its properties. In particular, we show that the solution exists for all real times and that it remains extremally Ricci-pinched. This result holds m…

Among closed G2-structures there are two very distinguished classes: Laplacian solitons and Extremally Ricci-pinched G2-structures. We study the existence problem and explore possible interplays between these concepts in the context of left-invariant G2-structures on solvable Lie groups. Also, some Ricci pinching prope…

Classification of G2-structures on Lie groups with Ricci pinched conditions.

The paper examines rigidity of special submanifolds in spheres with curvature constraints.

Three-manifolds with non-negative pinched Ricci curvature have complete Ricci flows.

We give the first examples of closed Laplacian solitons which are shrinking, and in particular produce closed Laplacian flow solutions with a finite-time singularity. Extremally Ricci pinched G2-structures (introduced by Bryant) which are steady Laplacian solitons have also been found. All the examples are left-invaria…

The paper pinches curvature in expanding Ricci solitons.

Alternative proof of flatness for Ricci-pinched 3-manifolds.

Ricci flow on flat manifolds converges to Euclidean space under curvature pinching.

New solitons found for G$_2$-Laplacian flow on Lie groups.

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.

This paper extends 3D results to higher dimensions, proving compactness for PIC1 pinched manifolds.

A 3-manifold's Ricci pinching condition implies it's flat if it has Euclidean volume growth.

In this paper, we proved a compactness result about Riemannian manifolds with an arbitrary pointwisely pinched Ricci curvature tensor.

The paper investigates quantitative rigidity using Colding's monotonicity formulas for Ricci curvature.

Proves pinched Ricci curvature conjecture in all dimensions.

Compact shrinkers with curvature pinching conditions proven.

We show that any ancient solution to the Ricci flow which satisfies a suitable curvature pinching condition must have constant sectional curvature.

We give examples of pinched negatively curved manifolds for which the Ricci flow does not converge smoothly.

New curvature condition proves rigidity of Bryant Ricci solitons.

New proof confirms noncompact locally conformally flat manifolds are compact.

Study new Ricci flow invariant curvature conditions.

We prove that a $n$-dimensional, $4 \leq n \leq 6$, compact gradient shrinking Ricci soliton satisfying a $L^{n/2}$-pinching condition is isometric to a quotient of the round $\mathbb{S}^{n}$. The proof relies mainly on sharp algebraic curvature estimates, the Yamabe-Sobolev inequality and an improved rigidity result f…

We study curvature pinching estimates of Ricci flow on complete 3- dimensional manifolds without bounded curvature assumption. We will derive some general curvature conditions which are preserved on any complete solution of 3-dim Ricci flow, these conditions include nonnegative Ricci curvature and sectional curvature a…

In earlier work, carrying out numerical simulations of the Ricci flow of families of rotationally symmetric geometries on $S3$, we have found strong support for the contention that (at least in the rotationally symmetric case) the Ricci flow for a ``critical'' initial geometry - one which is at the transition point bet…

In this note we consider versions of both Ricci and sectional curvature pinching for Riemannian manifold with density. In the Ricci curvature case the main result implies a diameter estimate that is new even for compact shrinking Ricci solitons. In the case of sectional curvature we prove a new sphere theorem.

Proves long-time Ricci flow existence and topological rigidity for pinched integral curvature manifolds.

Yamabe flow proves compactness of certain locally conformally flat manifolds with positive Ricci curvature.

In this paper, we give the full proof of a conjecture of R.Hamilton that for $(M^3, g)$ being a complete Riemannian 3-manifold with bounded curvature and with the Ricci pinching condition $Rc\geq \ep R g$, where $R>0$ is the positive scalar curvature and $\ep>0$ is a uniform constant, $M^3$ is compact. One of the key i…

We prove that an $n$-dimensional, $n\geq4$, compact gradient shrinking Ricci soliton satisfying a $L^{\frac n2}$-pinching condition is isometric to a quotient of the round $\mathbb{S}^n$, which improves the rigidity theorem given by G. Catino (arXiv:1509.07416vl).

We prove that any $n$--dimensional complete gradient Ricci soliton with pinched Weyl curvature is a finite quotient of $\RR^{n}$, $\RR \times \SS^{n-1}$ or $\SS^{n}$. In particular, we do not need to assume the metric to be locally conformally flat.

In this paper, we first derive a pinching estimate on the traceless Ricci curvature in term of scalar curvature and Weyl tensor under the Ricci flow. Then we apply this estimate to study finite-time singularity behavior. We show that if the scalar curvature is uniformly bounded, then the Weyl tensor has to blow up, as …

We prove pinching estimates for solutions of the linearized Ricci flow system on a closed manifold of dimension $n\geq 4$ with positive scalar curvature and vanishing Weyl tensor. If the vanishing Weyl tensor condition is removed, we only give a rough pinching estimate controlled by some blow-up function in a short tim…

The study characterizes compact submanifolds with pinched Ricci curvature in Euclidean and spherical space forms.

Considering the almost rigidity of the Obata theorem, we generalize Petersen and Aubry's sphere theorem about eigenvalue pinching without assuming the positivity of Ricci curvature, only assuming $Ric\geq-Kg$ and $diam\leq D$ for some positive constants $K>0$ and $D>0$.

New proof shows 3-manifolds with specific curvature properties are either flat or compact.

Study pinches curvature under Laplacian G_2 flow, proving Weyl tensor norm blows up.

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…

Researchers prove solvmanifolds are global maxima for Ricci pinching functional in new cases.

The abstract discusses compactness of manifolds with pinched Ricci curvature.

Found a new compact G2-structure on a 7-manifold.

Study approximates product of spheres using Laplacian eigenvalues.

Study on compact manifolds for exact G$_{2}$-Structures without additional constraints.

We prove an estimate for solutions to the linearized Ricci flow system on closed 3-manifolds. This estimate is a generalization of Hamilton's pinching is preserved estimate for the Ricci curvatures of solutions to the Ricci flow on 3-manifolds with positive Ricci curvature. In our estimate we make no assumption on the …

Let (M,g_0) be a compact Riemannian manifold with pointwise 1/4-pinched sectional curvatures. We show that the Ricci flow deforms g_0 to a constant curvature metric. The proof uses the fact, also established in this paper, that positive isotropic curvature is preserved by the Ricci flow in all dimensions. We also rely …

The paper proves stability of Ricci flow for certain initial conditions.

Study on 4D solitons with curvature constraints.