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Let $\varepsilon>0$ be sufficiently small. Denote by $\mathrm{Rm}$ and $\mathrm{R}$ the curvature operator and the scalar curvature.

Consider the following pinching condition $$\langle\mathrm{Rm}\,\phi,\phi\rangle\ge -\varepsilon\,\mathrm{R}\,|\phi|^2$$ for any tangent bivector $\phi$.

Questions

  1. Is this piching preserved under Ricci flow? (This is true in 3-dimensional case, see Bing-Long Chen, Guoyi Xu, Zhuhong Zhang, Local pinching estimates in 3-dim Ricci flow.)

  2. If yes, any references?

  3. If no, is there any cone of curvature operators which is preserved under Ricci flow and just little wider than the cone $\mathrm{Rm}\ge 0$.

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As far as I know, this question is not addressed anywhere.

However, I have to say I don't expect this precise estimate to hold. At least, it doesn't seem to follow from Hamilton's maximum principle.

The closest thing I know is the work of Miles Simon in dimension 3 (http://arxiv.org/abs/math/0612095). Lemma 4.1 looks like the kind of estimate you are looking for but with a time dependent $\varepsilon$ and an additive constant added in, for Ricci curvature in dimension 3. It is valid only for short time.

I actually tried to generalise Simon's estimate to the curvature operator in arbitrary dimensions. The closest thing I ended up with can be found here (http://arxiv.org/abs/1111.0859), this is Theorem 1.3. Unfortunately, it is really weaker than what you want.

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