Abstract
In this paper, we show that the eigenvalues of \((-\triangle+\frac{R}{2})\) are nondecreasing under the Ricci flow for manifolds with nonnegative curvature operator. Then we show that the only steady Ricci breather with nonnegative curvature operator is the trivial one which is Ricci-flat.
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References
Chow B. (1991): The Ricci flow on the 2-sphere. J. Differ. Geom. 33(2): 325–334
Hamilton R.S. (1982): Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17(2): 255–306
Hamilton R.S. (1986): Four-manifolds with positive curvature operator. J. Differential Geom. 24(2): 153–179
Hamilton, R.S.: The Ricci flow on surfaces. In: Mathematics and General Relativity (Santa Cruz, CA, 1986), pp. 237–262. American Mathematices Society, Providence (1988)
Ma, L.: Eigenvalue monotonicity for the ricci-hamilton flow, revised version (2005)
Perelman, G.: The entropy formula for the ricci flow and its geometric applications (2002)
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Cao, X. Eigenvalues of \((-\triangle + \frac{R}{2})\) on manifolds with nonnegative curvature operator. Math. Ann. 337, 435–441 (2007). https://doi.org/10.1007/s00208-006-0043-5
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DOI: https://doi.org/10.1007/s00208-006-0043-5