Mean curvature of direct image bundles

Let $E\to X$ be a vector bundle of rank $r$ over a compact complex manifold $X$ of dimension $n$. It is known that if the line bundle $O_{P(E^*)}(1)$ over the projectivized bundle $P(E^*)$ is positive, then $E\otimes \det E$ is Nakano positive by the work of Berndtsson. In this paper, we give a subharmonic analogue. Let $p:P(E^*)\to X$ be the projection and $\alpha$ be a K\"ahler form on $X$. If the line bundle $O_{P(E^*)}(1)$ admits a metric $h$ with curvature $\Theta$ positive on every fiber and $\Theta^r\wedge p^*\alpha^{n-1}> 0$, then $E\otimes \det E$ carries a Hermitian metric whose mean curvature is positive. As an application, we show that the following subharmonic analogue of the Griffiths conjecture is true: if the line bundle $O_{P(E^*)}(1)$ admits a metric $h$ with curvature $\Theta$ positive on every fiber and $\Theta^r\wedge p^*\alpha^{n-1}> 0$, then $E$ carries a Hermitian metric with positive mean curvature.