[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/JKMS.2013.50.4.829

A NEW 3-PARAMETER CURVATURE CONDITION PRESERVED BY RICCI FLOW

Gao, Xiang (School of Mathematical Sciences, Ocean University of China)

Publication Information

Journal of the Korean Mathematical Society / v.50, no.4, 2013 , pp. 829-845 More about this Journal

Abstract

In this paper, we firstly establish a family of curvature invariant conditions lying between the well-known 2-nonnegative curvature operator and nonnegative curvature operator along the Ricci flow. These conditions are defined by a set of inequalities involving the first four eigenvalues of the curvature operator, which are named as 3-parameter ${\lambda}$ -nonnegative curvature conditions. Then a related rigidity property of manifolds with 3-parameter ${\lambda}$ -nonnegative curvature operators is also derived. Based on these, we also obtain a strong maximum principle for the 3-parameter ${\lambda}$ -nonnegativity along Ricci flow.

Keywords

Ricci flow; 3-parameter ${\lambda}$ -nonnegative curvature operator; maximum principle;

Citations & Related Records

Reference

1	C. Bohm and B. Wilking, Manifolds with positive curvature operators are space forms, Ann. of Math. (2) 167 (2008), no. 3, 1079-1097. DOI
2	H. W. Chen, Pointwise 1/4-pinched 4-manifolds, Ann. Global Anal. Geom. 9 (1991), no. 2, 161-176. DOI
3	B. Chow, S. C. Chu, D. Glickenstein, C. Guentheretc, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo, and L. Ni, The Ricci flow: techniques and applications. Part II, Mathematical Surveys and Monographs, 144, AMS, Providence, RI, 2008.
4	B. Chow and P. Lu, The maximum principle for systems of parabolic equations subject to an avoidance set, Pacific J. Math. 214 (2004), no. 2, 201-222. DOI
5	B. Chow, P. Lu, and L. Ni, Hamilton's Ricci Flow, American Mathematical Society/Science Press, Providence, 2006.
6	R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1999.
7	M. J. Micallef and J. D. 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. DOI ScienceOn
8	M. J. Micallef and M. Y. Wang, Metrics with nonnegative isotropic curvature, Duke Math. J. 72 (1993), no. 3, 649-672. DOI