Characterizations of Griffiths Positivity, Pluriharmonicity and Flatness

Deng-Ning-Wang-Zhou showed that a Hermitian holomorphic vector bundle is Griffiths semi-positive if it satisfies the optimal $L^2$-extension condition. As a generalization, we present a quantitative characterization of Griffiths positivity in terms of certain $L^2$-extension conditions. We also show that a $\mathbb{R}$-valued measurable function is pluriharmonic if and only if it satisfies the equality part of the optimal $L^p$-extension condition. This answers a conjecture of Inayama affirmatively. Moreover, the flatness of a possibly singular Hermitian metric is also equivalent to the equality part of the optimal $L^p$-extension condition.