## Non-coercive Ricci flow invariant curvature cones

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- by Thomas Richard and Harish Seshadri PDF
- Proc. Amer. Math. Soc.
**143**(2015), 2661-2674 Request permission

## Abstract:

This note is a study of nonnegativity conditions on curvature preserved by the Ricci flow. We focus on a specific class of curvature conditions which we call non-coercive: These are the conditions for which nonnegative curvature and vanishing scalar curvature does not imply flatness.

We show, in dimensions greater than $4$, that if a Ricci flow invariant nonnegativity condition is satisfied by all Einstein curvature operators with nonnegative scalar curvature, then this condition is just the nonnegativity of scalar curvature. As a corollary, we obtain that a Ricci flow invariant curvature condition, which is stronger than a nonnegative scalar curvature, cannot be strictly satisfied by curvature operators (other than multiples of the identity) of compact Einstein symmetric spaces.

We also investigate conditions which are satisfied by all conformally flat manifolds with nonnegative scalar curvature.

## References

- Simon Brendle,
*Einstein manifolds with nonnegative isotropic curvature are locally symmetric*, Duke Math. J.**151**(2010), no. 1, 1–21. MR**2573825**, DOI 10.1215/00127094-2009-061 - Simon Brendle,
*Ricci flow and the sphere theorem*, Graduate Studies in Mathematics, vol. 111, American Mathematical Society, Providence, RI, 2010. MR**2583938**, DOI 10.1090/gsm/111 - Simon Brendle,
*Einstein metrics and preserved curvature conditions for the Ricci flow*, Complex and differential geometry, Springer Proc. Math., vol. 8, Springer, Heidelberg, 2011, pp. 81–85. MR**2964469**, DOI 10.1007/978-3-642-20300-8_{4} - Simon Brendle and Richard Schoen,
*Manifolds with $1/4$-pinched curvature are space forms*, J. Amer. Math. Soc.**22**(2009), no. 1, 287–307. MR**2449060**, DOI 10.1090/S0894-0347-08-00613-9 - Christoph Böhm and Burkhard Wilking,
*Manifolds with positive curvature operators are space forms*, Ann. of Math. (2)**167**(2008), no. 3, 1079–1097. MR**2415394**, DOI 10.4007/annals.2008.167.1079 - H. A. Gururaja, Soma Maity, and Harish Seshadri,
*On Wilking’s criterion for the Ricci flow*, Math. Z.**274**(2013), no. 1-2, 471–481. MR**3054339**, DOI 10.1007/s00209-012-1079-8 - M. Gromov,
*Sign and geometric meaning of curvature*, Rend. Sem. Mat. Fis. Milano**61**(1991), 9–123 (1994) (English, with English and Italian summaries). MR**1297501**, DOI 10.1007/BF02925201 - Richard S. Hamilton,
*Three-manifolds with positive Ricci curvature*, J. Differential Geometry**17**(1982), no. 2, 255–306. MR**664497** - Richard S. Hamilton,
*Four-manifolds with positive curvature operator*, J. Differential Geom.**24**(1986), no. 2, 153–179. MR**862046** - Mario J. Micallef and John Douglas Moore,
*Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes*, Ann. of Math. (2)**127**(1988), no. 1, 199–227. MR**924677**, DOI 10.2307/1971420 - Huy T. Nguyen,
*Isotropic curvature and the Ricci flow*, Int. Math. Res. Not. IMRN**3**(2010), 536–558. MR**2587576**, DOI 10.1093/imrn/rnp147 - Burkhard Wilking,
*A Lie algebraic approach to Ricci flow invariant curvature conditions and Harnack inequalities*, J. Reine Angew. Math.**679**(2013), 223–247. MR**3065160**, DOI 10.1515/crelle.2012.018

## Additional Information

**Thomas Richard**- Affiliation: Department of Mathematics, Indian Institute of Science, 560012 Bangalore, India
- Email: thomas@math.iisc.ernet.in, thomas.richard@u-pec.fr
**Harish Seshadri**- Affiliation: Department of Mathematics, Indian Institute of Science, 560012 Bangalore, India
- MR Author ID: 712201
- Email: harish@math.iisc.ernet.in
- Received by editor(s): August 15, 2013
- Received by editor(s) in revised form: January 8, 2014
- Published electronically: February 16, 2015
- Communicated by: Lei Ni
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**143**(2015), 2661-2674 - MSC (2010): Primary 53C44
- DOI: https://doi.org/10.1090/S0002-9939-2015-12619-6
- MathSciNet review: 3326045