arXiv research

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

The paper pinches curvature in expanding Ricci solitons.

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

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.

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…

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…

We prove a curvature pinching result for the Ricci flow on asymptotically flat manifolds: if an asymptotically flat manifold of dimension $n\geq 3$ has scale-invariant integral norm of curvature sufficiently pinched relative to the inverse of its Sobolev constant, then the Ricci flow starting from this manifold exists …

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…

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…

Only two examples of extremally Ricci pinched G2-structures can be found in the literature and they are both homogeneous. We study in this paper the existence and structure of such very special closed G2-structures on Lie groups. Strong structural conditions on the Lie algebra are proved to hold. As an application, we …

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).

There is a conjecture that a complete Riemannian 3-manifold with bounded sectional curvature, and pointwise pinched nonnegative Ricci curvature, must be flat or compact. We show that this is true when the negative part (if any) of the sectional curvature decays quadratically.

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 $D>0$.

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…

The abstract discusses compactness of manifolds with pinched Ricci curvature.

Study approximates product of spheres using Laplacian eigenvalues.

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.

Study rigidity of minimal Legendrian submanifolds in spheres via eigenvalues.

We examine a Type-1 neck pinch singularity in simplicial Ricci flow (SRF) for an axisymmetric piecewise flat 3-dimensional geometry with 3-sphere topology. SRF was recently introduced as an unstructured mesh formulation of Hamilton's Ricci flow (RF). It describes the RF of a piecewise-flat simplicial geometry. In this …

New classification for higher-dimensional shrinking Ricci solitons with positive isotropic curvature.

New rigidity results for critical metrics with curvature pinching.

We use numerical techniques to study the formation of singularities in Ricci flow. Comparing the Ricci flows corresponding to a one parameter family of initial geometries on S^3 with varying amounts of S^2 neck pinching, we find critical behavior at the threshold of singularity formation.

We prove that if the initial hypersurface of the mean curvature flow in spheres satisfies a sharp pinching condition, then the solution of the flow converges to a round point or a totally geodesic sphere. Our result improves the famous convergence theorem due to Huisken [9]. Moreover, we prove a convergence theorem und…

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