Recently I read the following results.

(1) Zheng, F.. *Kodaira dimensions and hyperbolicity of
nonpositively curved compact K\"ahler manifolds.* Comment. Math.
Helv. 77 (2002), no. 2, 221-234.

(2) Jahnke, P.; Peternell, T.; Radloff, I.. *Some recent
developments in the classification theory of higher dimensional
manifolds.* Global aspects of complex geometry, 311-357, Springer,
Berlin, 2006.

**Theorem** [Zheng Thm 2 p232] Let $M^2$ be a compact
K\"ahler surface of nonpositive sectional curvature. If it is of
general type, then it is Kobayashi hyperbolic.

**Theorem** [Jahnke-Peternell-Radloff Cor 6.10 p334] Let
$X_n$ be a projective manifold whose universal cover is Stein (or
has no positive-dimensional subvariety). Then either $K_X$ is ample
or $\chi(\mathcal{O}_X) = 0$, $K_X$ is nef and $K_X^n=0$.

It seems that both results rely on the following step:

**Proposition**: Any minimal surface or variety $X_n$ of
general type ($K_X$ big and nef), Assume $X_n$ is projective (no
need when $n=2$), then $K_X$ is ample if $X_n$ admits no rational
curve.

Both two papers refer this as a 1991 or 1992 result of Kawamata, however, I failed to find a precise statement. Can anyone give a precise statement of Kawamata's result or point out a reference?