1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

APPENDIX B Other, Aspects of Ricci Flow and Related Flows Convergence to Ricci solitons Given that convergence to a soliton pla ...

478 B. OTHER ASPECTS OF RICCI FLOW AND RELATED FLOWS THEOREM B.2 (Wu [373]). Let go be a complete metric on lR^2 , of the form ( ...

CONVERGENCE TO RICCI SOLITONS 479 the derivative of the conformal factor for g give subsequence convergence, on any compact se ...

480 B. OTHER ASPECTS OF RICCI FLOW AND RELATED FLOWS In higher dimensions, there are not many results. However, consider complet ...

CONVERGENCE TO RICCI SOLITONS 481 However' the time-derivative a I at under the Ricci fl.ow and the radial- derivative a/ ar d ...

482 B. OTHER ASPECTS OF RICCI FLOW AND RELATED FLOWS 2. The mean curvature fl.ow In this section we give a brief introduction to ...

THE MEAN CURVATURE FLOW 483 Note that H = gij hij and (B.12) We have the following basic formulas for solutions of the mean cu ...

484 B. OTHER ASPECTS OF RICCI FLOW AND RELATED FLOWS evolution of the metric is given by :t9ij = ( D ~; ( aa~) , ~~) + ( ~~, D ~ ...

THE MEAN CURVATURE FLOW 485 Tracing the above formula, we see that the mean curvature evolves by aH a .. a at = -at9ij · hij + ...

486 B. OTHER ASPECTS OF RICCI FLOW AND RELATED FLOWS Thus, considering Rep as a 2-tensor on P and changing its covariant derivat ...

THE MEAN CURVATURE FLOW 487 EXERCISE B.9. Compute gt (lhJ^2 - ~H^2 ). See §5 of Huisken [212] for a study of under what condit ...

488 B. OTHER ASPECTS OF RICCI FLOW AND RELATED FLOWS Using the facts that (B.28) where XT ~ X - (X, v) v (.6.. is the Laplacian ...

THE MEAN CURVATURE FLOW 489 Formula (B.26) is useful for studying Type I singularities of the mean curvature fl.ow, where sup ...

490 B. OTHER ASPECTS OF RICCI FLOW AND RELATED FLOWS matrix Harnack inequality for the heat equation (see Part II of this volume ...

THE CROSS CURVATURE FLOW 491 3.1. The cross curvature tensor. Let (M^3 ,g) be a Riemannian 3- manifold with either negative se ...

492 B. OTHER ASPECTS OF RICCI FLOW AND RELATED FLOWS and similarly for a22 and a33. D A nice property of the cross curvature ten ...

THE CROSS CURVATURE FLOW 493 THEOREM B.21 (XCF short-time existence). If (M^3 , g 0 ) is a closed 3- manifold with either nega ...

494 B. OTHER ASPECTS OF RICCI FLOW AND RELATED FLOWS Hence, if we take ei ® ei, ei ® e2 + e2 ® ei, ei ® e3 + e3 ® ei, e2 ® e2, e ...

THE CROSS CURVATURE FLOW Hence [a-DY (g) (() v]ij = gpq ((i(pVqj + (j(pVqi - (i(jVpq) = 8i1V1j + Oj1V1i - 8il8j1gpqVpq, assumi ...

496 B. OTHER ASPECTS OF RICCI FLOW AND RELATED FLOWS REMARK B.22. The above proof follows [34], where it is pointed out that the ...