$U(n)$-invariant Kähler–Ricci flow with non-negative curvature

Abstract

It is interesting to know the existence of the Kähler-Ricci flow on complete non-compact Kähler manifolds with non-negative holomorphic bisectional curvature. In this paper, we study U (n)invariant Kähler-Ricci flow on C n with non-negative curvature. Motivated by the recent work of Wu and the second named author [37], we also study examples of U (n)-invariant complete Kähler metrics on C n with positive and unbounded curvature.252 Bo Yang & Fangyang Zheng 4 Discussions on the existence of U (n)-invariant Kähler-Ricci flow 283 4.1 Theorem of Cabezas-Rivas and Wilking 283 4.2 Ricci flow on double covers with rotational symmetry 283 4.3 Conditions that ensure the existence of Kähler-Ricci flow 285 Acknowledgements 290 References 290

Gap theorem on Kähler manifolds with nonnegative orthogonal bisectional curvature

“…In [21], Wu and Zheng considered the U(n)-invariant Kähler metrics on C n and obtained necessary and sufficient conditions for the nonnegativity of the curvature operator, nonnegativity of the sectional curvature, as well as the nonnegativity of the bisectional curvature respectively. In [22], Yang and Zheng later proved that the necessary and sufficient condition in [21] for the nonnegativity of the sectional curvature holds for the nonnegativity of the complex sectional curvature under the unitary symmetry. In [7], the authors obtained the necessary and sufficient conditions for (NOB) and (NQOB) respectively.…”

“…We follow the same notations as in [21,22]. Let (z 1 , · · · , z n ) be the standard coordinate on C n and r = |z| 2 .…”

Comparison and vanishing theorems for Kähler manifolds

“…We follow the same notations as in [36,38]. Let (z 1 , · · · , z m ) be the standard coordinate on C m and r = |z| 2 .…”

Longtime existence of the Kähler-Ricci flow on ℂⁿ

“…We may assume that ξ is not identically zero, otherwise g 0 would be the standard metric and the theorem is obviously true. By [3,Theoerm 5.4] and [24], the Kähler-Ricci flow has a U(n)invariant complete solution g(t) on M × [0, T ) which is equivalent to g 0 and has bounded non-negative bisectional curvature. Hence we may assume that g 0 has bounded curvature.…”

“…The short time existence of g(t) in the above Theorem was proved by Chau-Li-Tam in [3], while the fact that the solution has non-negative bisectional curvature was proved by Yang-Zheng in [24]. The existence of complete U(n)-invariant Kähler metrics with unbounded nonnegative bisectional curvature was shown in [22] and thus, in the U(n)invariant case, our results extend the longtime existence results of Shi [16].…”