Abstract
AbstractThis article aims to understand the behavior of the curvature operator of the second kind under the Ricci flow in dimension three. First, we express the eigenvalues of the curvature operator of the second kind explicitly in terms of that of the curvature operator (of the first kind). Second, we prove that $$\alpha $$
α
-positive/$$\alpha $$
α
-nonnegative curvature operator of the second kind is preserved by the Ricci flow in dimension three for all $$\alpha \in [1,5]$$
α
∈
[
1
,
5
]
.
Funder
Simons Foundation
National Science Foundation
Publisher
Springer Science and Business Media LLC
Reference50 articles.
1. Andrews, B., Nguyen, H.: Four-manifolds with 1/4-pinched flag curvatures. Asian J. Math. 13(2), 251–270 (2009)
2. Berger, M., Ebin, D.: Some decompositions of the space of symmetric tensors on a Riemannian manifold. J. Differ. Geom. 3, 379–392 (1969)
3. Besse, A.L.: Einstein Manifolds. Classics in Mathematics. Springer, Berlin (2008). (Reprint of the 1987 edition)
4. Bourguignon, J.-P., Karcher, H.: Curvature operators: pinching estimates and geometric examples. Ann. Sci. École Norm. Sup. (4) 11(1), 71–92 (1978)
5. Borel, A.: On the curvature tensor of the Hermitian symmetric manifolds. Ann. Math. 2(71), 508–521 (1960)