On the Chern-type problem in an indefinite Kähler geometry (Q5939526)

The authors prove the following theorem: Let \(M\) be an \(n(\geq 3)\)-dimensional complete space-like complex submanifold of an \((n+p)\)-dimensional indefinite complex hyperbolic space \(\mathbb{C} H_p^{n+p}(c)\) of constant holomorphic sectional curvature \(c\) and of index \(2p(>0)\). If \(M\) is not totally geodesic and \(p\leq(1/2)n (n+1)\), then the squared norm \(h_2\) of the second fundamental form \(\alpha\) of \(M\) satisfies NEWLINE\[NEWLINEh_2\geq{cnp (n+2)\over 2(n+2p)},NEWLINE\]NEWLINE where the equality holds if and only if \(M\) is a complex projective space \(\mathbb{C} P^n (c/2)\), \(\alpha\) is parallel and \(p=(1/2)n(n+1)\).