# The Ricci flow under almost non-negative curvature conditions

@article{Bamler2017TheRF, title={The Ricci flow under almost non-negative curvature conditions}, author={Richard Bamler and Esther Cabezas-Rivas and Burkhard Wilking}, journal={Inventiones mathematicae}, year={2017}, volume={217}, pages={95-126} }

We generalize most of the known Ricci flow invariant non-negative curvature conditions to less restrictive negative bounds that remain sufficiently controlled for a short time. As an illustration of the contents of the paper, we prove that metrics whose curvature operator has eigenvalues greater than $$-\,1$$-1 can be evolved by the Ricci flow for some uniform time such that the eigenvalues of the curvature operator remain greater than $$-\,C$$-C. Here the time of existence and the constant C… Expand

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#### References

SHOWING 1-10 OF 34 REFERENCES

Ricci flow under local almost non-negative curvature conditions

- Mathematics
- 2018

We find a local solution to the Ricci flow equation under a negative lower bound for many known curvature conditions. The flow exists for a uniform amount of time, during which the curvature stays… Expand

The entropy formula for the Ricci flow and its geometric applications

- Mathematics
- 2002

We present a monotonic expression for the Ricci flow, valid in all dimensions and without curvature assumptions. It is interpreted as an entropy for a certain canonical ensemble. Several geometric… Expand

Ricci flow of almost non-negatively curved three manifolds

- Mathematics
- 2006

Abstract In this paper we study the evolution of almost non-negatively curved (possibly singular) three dimensional metric spaces by Ricci flow. The non-negatively curved metric spaces which we… Expand

A Lie algebraic approach to Ricci flow invariant curvature conditions and Harnack inequalities

- Mathematics
- 2010

Abstract We consider a subset S of the complex Lie algebra 𝔰𝔬(n, ℂ) and the cone C(S) of curvature operators which are nonnegative on S. We show that C(S) defines a Ricci flow invariant… Expand

A Lie algebraic approach to Ricci flow invariant curvature conditions and Harnack inequalities

- Mathematics
- 2013

We consider a subset S of the complex Lie algebra soðn;CÞ and the cone CðSÞ of curvature operators which are nonnegative on S. We show that CðSÞ defines a Ricci flow invariant curvature condition if… Expand

How to produce a Ricci Flow via Cheeger-Gromoll exhaustion

- Mathematics
- 2011

We prove short time existence for the Ricci flow on open manifolds of nonnegative complex sectional curvature. We do not require upper curvature bounds. By considering the doubling of convex sets… Expand

Ricci flow from spaces with isolated conical singularities

- Mathematics
- 2016

Let $(M,g_0)$ be a compact $n$-dimensional Riemannian manifold with a finite number of singular points, where the metric is asymptotic to a non-negatively curved cone over $(\mathbb{S}^{n-1},g)$. We… Expand

An Introduction to the Kähler–Ricci Flow

- Mathematics
- 2013

These notes give an introduction to the Kahler–Ricci flow. We give an exposition of a number of well-known results including: maximal existence time for the flow, convergence on manifolds with… Expand

Ricci flow of non-collapsed three manifolds whose Ricci curvature is bounded from below

- Mathematics
- 2009

Abstract We consider complete (possibly non-compact) three dimensional Riemannian manifolds (M, g) such that: (a) (M, g) is non-collapsed (i.e. the volume of an arbitrary ball of radius one is… Expand

Manifolds with positive curvature operators are space forms

- Mathematics
- 2006

The Ricci flow was introduced by Hamilton in 1982 [H1] in order to prove that a compact three-manifold admitting a Riemannian metric of positive Ricci curvature is a spherical space form. In… Expand