##
**Ricci flow with surgery on manifolds with positive isotropic curvature.**
*(English)*
Zbl 1423.53080

Summary: We study the Ricci flow for initial metrics with positive isotropic curvature (strictly PIC for short).

In the first part of this paper, we prove new curvature pinching estimates that ensure that blow-up limits are uniformly PIC in all dimensions. Moreover, in dimension \(n\geq 12\), we show that blow-up limits are weakly \(\mathrm{PIC}2\). This can be viewed as a higher-dimensional version of the fundamental Hamilton-Ivey pinching estimate in dimension \(3\).

In the second part, we develop a theory of ancient solutions that have bounded curvature, are \(\kappa\)-noncollapsed, are weakly \(\mathrm{PIC}2\), and are uniformly PIC. This is an extension of Perelman’s work; the additional ingredients needed in the higher dimensional setting are the differential Harnack inequality for solutions to the Ricci flow satisfying the PIC2 condition, and a rigidity result due to Brendle-Huisken-Sinestrari for ancient solutions that are uniformly \(\mathrm{PIC}1\).

In the third part of this paper, we prove a Canonical Neighborhood Theorem for the Ricci flow with initial data with positive isotropic curvature, which holds in dimension \(n \geq 12\). This relies on the curvature pinching estimates together with the structure theory for ancient solutions. This allows us to adapt Perelman’s surgery procedure to this situation. As a corollary, we obtain a topological classification of all compact manifolds with positive isotropic curvature of dimension \(n \geq 12\) that do not contain nontrivial incompressible \((n-1)\)-dimensional space forms.

In the first part of this paper, we prove new curvature pinching estimates that ensure that blow-up limits are uniformly PIC in all dimensions. Moreover, in dimension \(n\geq 12\), we show that blow-up limits are weakly \(\mathrm{PIC}2\). This can be viewed as a higher-dimensional version of the fundamental Hamilton-Ivey pinching estimate in dimension \(3\).

In the second part, we develop a theory of ancient solutions that have bounded curvature, are \(\kappa\)-noncollapsed, are weakly \(\mathrm{PIC}2\), and are uniformly PIC. This is an extension of Perelman’s work; the additional ingredients needed in the higher dimensional setting are the differential Harnack inequality for solutions to the Ricci flow satisfying the PIC2 condition, and a rigidity result due to Brendle-Huisken-Sinestrari for ancient solutions that are uniformly \(\mathrm{PIC}1\).

In the third part of this paper, we prove a Canonical Neighborhood Theorem for the Ricci flow with initial data with positive isotropic curvature, which holds in dimension \(n \geq 12\). This relies on the curvature pinching estimates together with the structure theory for ancient solutions. This allows us to adapt Perelman’s surgery procedure to this situation. As a corollary, we obtain a topological classification of all compact manifolds with positive isotropic curvature of dimension \(n \geq 12\) that do not contain nontrivial incompressible \((n-1)\)-dimensional space forms.

### MSC:

53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |

### References:

[1] | B\"{o}hm, Christoph; Wilking, Burkhard, Manifolds with positive curvature operators are space forms, Ann. of Math. (2). Annals of Mathematics. Second Series, 167, 1079-1097, (2008) · Zbl 1185.53073 |

[2] | Brendle, Simon, A general convergence result for the {R}icci flow in higher dimensions, Duke Math. J.. Duke Mathematical Journal, 145, 585-601, (2008) · Zbl 1161.53052 |

[3] | Brendle, Simon, A generalization of {H}amilton’s differential {H}arnack inequality for the {R}icci flow, J. Differential Geom.. Journal of Differential Geometry, 82, 207-227, (2009) · Zbl 1169.53050 |

[4] | Brendle, Simon, Einstein manifolds with nonnegative isotropic curvature are locally symmetric, Duke Math. J.. Duke Mathematical Journal, 151, 1-21, (2010) · Zbl 1189.53042 |

[5] | Brendle, Simon, Ricci Flow and the Sphere Theorem, Grad. Stud. Math., 111, viii+176 pp., (2010) · Zbl 1196.53001 |

[6] | Brendle, Simon, Ricci flow with surgery in higher dimensions, Ann. of Math. (2). Annals of Mathematics. Second Series, 187, 263-299, (2018) · Zbl 1393.53055 |

[7] | Brendle, Simon; Huisken, Gerhard; Sinestrari, Carlo, Ancient solutions to the {R}icci flow with pinched curvature, Duke Math. J.. Duke Mathematical Journal, 158, 537-551, (2011) · Zbl 1219.53062 |

[8] | Brendle, Simon; Schoen, Richard, Manifolds with {\(1/4\)}-pinched curvature are space forms, J. Amer. Math. Soc.. Journal of the American Mathematical Society, 22, 287-307, (2009) · Zbl 1251.53021 |

[9] | Cheeger, Jeff; Gromoll, Detlef, The splitting theorem for manifolds of nonnegative {R}icci curvature, J. Differential Geometry. Journal of Differential Geometry, 6, 119-128, (1971/72) · Zbl 0223.53033 |

[10] | Cheeger, Jeff; Gromoll, Detlef, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. (2). Annals of Mathematics. Second Series, 96, 413-443, (1972) · Zbl 0246.53049 |

[11] | Chen, Bing-Long; Zhu, Xi-Ping, Ricci flow with surgery on four-manifolds with positive isotropic curvature, J. Differential Geom.. Journal of Differential Geometry, 74, 177-264, (2006) · Zbl 1103.53036 |

[12] | Chen, Bing-Long; Tang, Siu-Hung; Zhu, Xi-Ping, Complete classification of compact four-manifolds with positive isotropic curvature, J. Differential Geom.. Journal of Differential Geometry, 91, 41-80, (2012) · Zbl 1257.53053 |

[13] | Chow, Bennett; Lu, Peng, The maximum principle for systems of parabolic equations subject to an avoidance set, Pacific J. Math.. Pacific Journal of Mathematics, 214, 201-222, (2004) · Zbl 1049.35101 |

[14] | Chow, Bennett; Chu, Sun-Chin; Glickenstein, David; Guenther, Christine; Isenberg, James; Ivey, Tom; Knopf, Dan; Lu, Peng; Luo, Feng; Ni, Lei, The {R}icci Flow: Techniques and Applications. {P}art {III}. {G}eometric-Analytic Aspects, Math. Surveys Monogr., 163, xx+517 pp., (2010) · Zbl 1216.53057 |

[15] | Fraser, Ailana M., Fundamental groups of manifolds with positive isotropic curvature, Ann. of Math. (2). Annals of Mathematics. Second Series, 158, 345-354, (2003) · Zbl 1044.53023 |

[16] | Gromov, M., Positive curvature, macroscopic dimension, spectral gaps and higher signatures. Functional Analysis on the eve of the 21st Century, {V}ol. {II}, Progr. Math., 132, 1-213, (1996) · Zbl 0945.53022 |

[17] | Hamilton, Richard S., Three-manifolds with positive {R}icci curvature, J. Differential Geom.. Journal of Differential Geometry, 17, 255-306, (1982) · Zbl 0504.53034 |

[18] | Hamilton, Richard S., Four-manifolds with positive curvature operator, J. Differential Geom.. Journal of Differential Geometry, 24, 153-179, (1986) · Zbl 0628.53042 |

[19] | Hamilton, Richard S., The {H}arnack estimate for the {R}icci flow, J. Differential Geom.. Journal of Differential Geometry, 37, 225-243, (1993) · Zbl 0804.53023 |

[20] | Hamilton, Richard S., The formation of singularities in the {R}icci flow. Surveys in Differential Geometry, {V}ol. {II}, 7-136, (1995) · Zbl 0867.53030 |

[21] | Hamilton, Richard S., Four-manifolds with positive isotropic curvature, Comm. Anal. Geom.. Communications in Analysis and Geometry, 5, 1-92, (1997) · Zbl 0892.53018 |

[22] | Huisken, Gerhard, Ricci deformation of the metric on a {R}iemannian manifold, J. Differential Geom.. Journal of Differential Geometry, 21, 47-62, (1985) · Zbl 0606.53026 |

[23] | Huisken, Gerhard; Sinestrari, Carlo, Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta Math.. Acta Mathematica, 183, 45-70, (1999) · Zbl 0992.53051 |

[24] | Huisken, Gerhard; Sinestrari, Carlo, Mean curvature flow with surgeries of two-convex hypersurfaces, Invent. Math.. Inventiones Mathematicae, 175, 137-221, (2009) · Zbl 1170.53042 |

[25] | Ivey, Thomas, Ricci solitons on compact three-manifolds, Differential Geom. Appl.. Differential Geometry and its Applications, 3, 301-307, (1993) · Zbl 0788.53034 |

[26] | Margerin, Christophe, Pointwise pinched manifolds are space forms. Geometric Measure Theory and the Calculus of Variations, Proc. Sympos. Pure Math., 44, 307-328, (1986) · Zbl 0587.53042 |

[27] | Margerin, Christophe, A sharp characterization of the smooth {\(4\)}-sphere in curvature terms, Comm. Anal. Geom.. Communications in Analysis and Geometry, 6, 21-65, (1998) · Zbl 0966.53022 |

[28] | Margerin, Christophe, D\'eformations de structures {R}iemanniennes · Zbl 0809.53042 |

[29] | Micallef, Mario J.; Moore, John Douglas, Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes, Ann. of Math. (2). Annals of Mathematics. Second Series, 127, 199-227, (1988) · Zbl 0661.53027 |

[30] | Micallef, Mario J.; Wang, McKenzie Y., Metrics with nonnegative isotropic curvature, Duke Math. J.. Duke Mathematical Journal, 72, 649-672, (1993) · Zbl 0804.53058 |

[31] | Morgan, John; Tian, Gang, Ricci Flow and the {P}oincar\'{e} Conjecture, Clay Math. Monogr., 3, xlii+521 pp., (2007) · Zbl 1179.57045 |

[32] | Nishikawa, Seiki, Deformation of {R}iemannian metrics and manifolds with bounded curvature ratios. Geometric Measure Theory and the Calculus of Variations, Proc. Sympos. Pure Math., 44, 343-352, (1986) · Zbl 0589.53046 |

[33] | Perelman, G., The entropy formula for the {R}icci flow and its geometric applications, (2002) · Zbl 1130.53001 |

[34] | Perelman, G., Ricci flow with surgery on three-manifolds, (2003) · Zbl 1130.53002 |

[35] | Perelman, G., Finite extinction time for solutions to the {R}icci flow on certain three-manifolds, (2003) · Zbl 1130.53003 |

[36] | Wilking, Burkhard, A {L}ie algebraic approach to {R}icci flow invariant curvature conditions and {H}arnack inequalities, J. Reine Angew. Math.. Journal f\`“{u}r die Reine und Angewandte Mathematik. [Crelle”s Journal], 679, 223-247, (2013) · Zbl 1380.53078 |

[37] | Yokota, Takumi, Complete ancient solutions to the {R}icci flow with pinched curvature, Comm. Anal. Geom.. Communications in Analysis and Geometry, 25, 485-506, (2017) · Zbl 1380.53079 |

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