# Remarks on the Extension of the Ricci Flow

@article{He2012RemarksOT, title={Remarks on the Extension of the Ricci Flow}, author={Fei He}, journal={Journal of Geometric Analysis}, year={2012}, volume={24}, pages={81-91} }

We present two new conditions to extend the Ricci flow on a compact manifold over a finite time, which are improvements of some known extension theorems.

## 10 Citations

### NOTES ON THE EXTENSION OF THE MEAN CURVATURE FLOW

- Mathematics
- 2014

In this paper, we present several new curvature conditions that assure the extension of the mean curvature flow on a finite time interval, which improve some known extension theorems.

### Mixed Integral Norms for Ricci Flow

- MathematicsThe Journal of Geometric Analysis
- 2020

We prove that a Ricci flow cannot develop a finite time singularity assuming the boundedness of a suitable space-time integral norm of the curvature tensor. Moreover, the extensibility of the flow is…

### Mean Value Inequalities and Conditions to Extend Ricci Flow

- Mathematics
- 2013

This paper concerns conditions related to the rst nite singularity time of a Ricci ow solution on a closed manifold. In particular, we provide a systematic approach to the mean value inequality…

### Ricci flow with bounded curvature integrals

- MathematicsPacific Journal of Mathematics
- 2021

In this paper, we study the Ricci flow on a closed manifold and finite time interval [0, T ) (T < ∞) on which certain integral curvature energies are finite. We prove that in dimension four, such…

### Type of finite time singularities of the Ricci flow with bounded scalar curvature

- Mathematics, Philosophy
- 2021

In this paper, we study the Ricci ﬂow on a closed manifold of dimension n ≥ 4 and ﬁnite time interval [0 , T ) ( T < ∞ ) on which the scalar curvature are uniformly bounded. We prove that if such ﬂow…

### A local singularity analysis for the Ricci flow and its applications to Ricci flows with bounded scalar curvature

- MathematicsCalculus of variations and partial differential equations
- 2022

We develop a refined singularity analysis for the Ricci flow by investigating curvature blow-up rates locally. We first introduce general definitions of Type I and Type II singular points and show…

## References

SHOWING 1-10 OF 27 REFERENCES

### Scalar Curvature Behavior for Finite Time Singularity of K

- Mathematics
- 2009

In this short paper, we show that K\"ahler-Ricci flows over closed manifolds would have scalar curvature blown-up for finite time singularity. Certain control of the blowing-up is achieved with some…

### The Ricci Flow: An Introduction

- Mathematics
- 2004

The Ricci flow of special geometries Special and limit solutions Short time existence Maximum principles The Ricci flow on surfaces Three-manifolds of positive Ricci curvature Derivative estimates…

### Curvature estimates for the Ricci flow II

- Mathematics
- 2005

In this paper we present several curvature estimates and convergence results for solutions of the Ricci flow, including the volume normalized Ricci flow and the normalized Kähler-Ricci flow. The…

### Curvature estimates for the Ricci flow I

- Mathematics
- 2008

In this paper we present several curvature estimates for solutions of the Ricci flow and the modified Ricci flow (including the volume normalized Ricci flow and the normalized Kähler-Ricci flow),…

### Best constant in Sobolev inequality

- Mathematics, Materials Science
- 1976

SummaryThe best constant for the simplest Sobolev inequality is exhibited. The proof is accomplished by symmetrizations (rearrangements in the sense of Hardy-Littlewood) and one-dimensional calculus…

### On type-I singularities in Ricci flow

- Mathematics
- 2011

We dene several notions of singular set for Type I Ricci ows and show that they all coincide. In order to do this, we prove that blow-ups around singular points converge to nontrivial gradient…

### Nonlinear analysis on manifolds: Sobolev spaces and inequalities

- Mathematics
- 1999

Elements of Riemannian geometry Sobolev spaces: The compact setting Sobolev spaces: The noncompact setting Best constants in the compact setting I Best constants in the compact setting II Optimal…

### On the Conditions to Extend Ricci Flow

- Mathematics
- 2008

Consider {(M n , g(t)), 0 ⩽ t < T < ∞} as an unnormalized Ricci flow solution: for t ∈ [0, T). Richard Hamilton shows that if the curvature operator is uniformly bounded under the flow for all t ∈…

### Remarks on the curvature behavior at the first singular time of the Ricci flow

- Mathematics
- 2010

@@ t gi jD 2Ri j; t2T0; T/; on a smooth, compact n-dimensional Riemannian manifold M. If the flow has uniformly bounded scalar curvature and develops Type I singularities at T , we show that suitable…

### On the conditions to control curvature tensors of Ricci flow

- Mathematics
- 2010

An important problem in the study of Ricci flow is to find the weakest conditions that provide control of the norm of the full Riemannian curvature tensor. In this article, supposing (Mn, g(t)) is a…