Asian Journal of Mathematics

Volume 19 (2015)

Number 1

$U(n)$-invariant Kähler metrics with nonnegative quadratic bisectional curvature

Pages: 1 – 16

DOI: http://dx.doi.org/10.4310/AJM.2015.v19.n1.a1

Authors

Shaochuang Huang (The Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong)

Luen-Fai Tam (The Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong)

Abstract

By perturbing the complete $U(n)$-invariant metrics with positive bisectional curvature constructed by Wu-Zheng [10], we obtain complete $U(n)$-invariant Kähler metrics on $\mathbb{C}^n , n \geq 3$, which have nonnegative quadratic bisectional curvature $(\mathbf{QB} \geq 0)$ everywhere, and which do not have nonnegative orthogonal bisectional curvature and do not have nonnegative Ricci curvature at some points. We prove that $\mathbf{QB} \geq 0$ is preserved under the Kähler-Ricci flow for complete $U(n)$-invariant solution with bounded curvature. We prove that $\mathrm{Ric} \geq 0$ is also preserved under an additional assumption.

Keywords

$U(n)$-invariant Kähler metrics, quadratic bisectional curvature, Kähler-Ricci flow

2010 Mathematics Subject Classification

Primary 32Q15. Secondary 53C44.

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Published 12 February 2015