K\"{a}hler manifolds with an almost $1/4$-pinched metric

Abstract

In this paper we construct an almost negatively $1/4$-pinched Riemannian metric on a class of compact manifolds recently discovered by Stover and Toledo in [17]. It is known that these manifolds are K\"{a}hler and not locally symmetric. These are the first known examples of not locally symmetric K\"{a}hler manifolds admitting such a metric and, via the result of Hernandez [9] and Yau and Zheng [18], these manifolds cannot admit a negatively quarter-pinched Riemannian metric. This metric is also interesting because it is a generalization to the complex hyperbolic setting of the famous pinched metric constructed by Gromov and Thurston in [8].