Negative sectional curvature and the product complex structure

Abstract

We prove that a product complex manifold cannot admit a complete Kähler metric with sectional curvature $K<c<0$ and Ricci curvature $Ric>d$, where $c$ and $d$ are constants. In particular, a product domain in $\C$ cannot cover a compact Kähler manifold with negative sectional curvature. On the other hand, we observe that there are complete Kähler metrics with negative sectional curvature on $\C$. Hence the upper sectional curvature bound is necessary.

6 Pages. To appear in Mathematical Research Letters