arXiv research

We refine a metric bunching estimate for pinched manifolds.

The paper pinches curvature in expanding Ricci solitons.

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

We prove pinching estimates for dual flows provided the curvature function used in the inverse flow in de Sitter space is convex.

We give new estimates for the extrinsic radius of compact hypersurfaces of the Euclidean space and the open hemisphere in terms of high order mean curvatures. Then we prove pinching results corresponding to theses estimates. We show that under a suitable pinching condition, the hypersurface is diffeomorphic and almost …

Sharp curvature estimates for mean curvature flow in spheres.

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…

Proves planarity and convexity for ancient solutions of mean curvature flow.

Sharp curvature pinching for mean curvature flow in spheres proved.

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…

Sharp estimates link curvature to topology, proving manifold rigidity.

We prove a sharp pinching estimate for immersed mean convex solutions of mean curvature flow which unifies and improves all previously known pinching estimates, including the umbilic estimate of Huisken, the convexity estimates of Huisken--Sinestrari and the cylindrical estimate of Huisken--Sinestrari. Namely, we show …

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…

Develops local curvature estimates for mean curvature flow.

The study pinches the rigidity of self-shrinking surfaces in mean curvature flow.

Study high codimension mean curvature flow in Riemannian manifolds, proving limiting flow in Euclidean space.

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

Second derivative pinching estimates are proved for a class of elliptic and parabolic equations, including motion of hypersurfaces by curvature functions such as quotients of elementary symmetric functions of curvature. The estimates imply convergence of convex hypersurfaces to spheres under these flows, improving earl…

We show that the blow-ups of compact solutions to the mean curvature flow in $\mathbb{R}^N$ initially satisfying the pinching condition $|A|^2 < c |H|^2$ for a suitable constant $c = c(n)$ must be codimension one.

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 …

Proves pinched Ricci curvature conjecture in all dimensions.

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.

For a convex domain $D$ bounded by the hypersurface $\partial D$ in a space of constant curvature we give sharp bounds on the width $R-r$ of a spherical shell with radii $R$ and $r$ that can enclose $\partial D$, provided that normal curvatures of $\partial D$ are pinched by two positive constants. Furthermore, in the …

Study mean curvature flow of high codimension submanifolds in complex projective space.

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 …

New findings on curvature and null spaces of Laplacians.

The paper studies hypersurfaces in spheres using mean curvature flow with surgery.

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…

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…

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

Study pinched self-dual Weyl curvature in compact 4-manifolds.

Paper proves Hamilton's pinching theorem using mean curvature flow.

The study pinches self-shrinking hypersurfaces in Euclidean space.

Flat Yang-Mills connections on pinched manifolds.

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

The paper establishes eigenvalue inequalities for a specific operator on curved spaces.

Compact shrinkers with curvature pinching conditions proven.

We prove new pinching estimate for the inverse curvature flow of strictly convex hypersurfaces in the space form $N$ of constant sectional curvature $K_N$ with speed given by $F^{-α}$, where $α\in (0,1]$ for $K_N=0,-1$ and $α=1$ for $K_N=1$, $F$ is a smooth, symmetric homogeneous of degree one function which is inverse…

Paper proves pinching theorem for minimal surfaces in spheres.

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

Proves CLT for Brownian paths on pinched negative curvature manifolds.

In this article, we generalize the classical Bochner-Weitzenböck theorem for manifolds satisfying an integral pinching on the curvature. We obtain the vanishing of Betti numbers under integral pinching assumptions on the curvature, and characterize the equality case. In particular, we reprove and extend to higher degre…

Sharp estimate for genus of embedded surfaces in 3-sphere.

Study pinched submanifolds in space forms, proving rigidity results.

We derive some elliptic differential inequalities from the Weitzenböck formulas for the traceless Ricci tensor of a Kähler manifold with constant scalar curvature and the Bochner tensor of a Kähler-Einstein manifold respectively. Using elliptic estimates and maximum principle, some $L^p$ and $L^\infty$ pinching result…

Harmonic maps between pinched Hadamard surfaces are quasi-conformal.

Study neck pinches in Lagrangian flows, proving stability and introducing new singularities.

Study on surfaces pinched by curvature in space forms converging under specific conditions.