Let $\mathcal E\to X$ be a stable vector bundle over a polarized projective manifold $(X,\omega)$.We know, that in this case $\mathcal E$ admits Hermitian-Einstein metric, i.e., a metric $h$, such that (may be up to some constants)
$$ {\rm tr}_\omega F(\mathcal E, h)=\lambda \rm Id, $$ where
- $F(\mathcal E, h)\in \Lambda^{1,1}(T^*M)\otimes \rm End(\mathcal E)$ is the curvature of Chern connection on $(\mathcal E, h)$
- ${\rm tr}_\omega\colon \Lambda^{1,1}(T^*M)\otimes \rm End(\mathcal E)\to \rm End(\mathcal E)$ is a contraction with $\omega$
- $\lambda=\int_X c_1(\mathcal E)\wedge\omega^{n-1}/\int_X \omega^n$.
Let $\pi : X\to\Delta$ be a degeneration of Hermitian-Einstein metrics $(X_s,\mathcal E_s) $ , then
$$ {\rm tr}_{\omega_s} F(\mathcal E_s, h_s)=\lambda(s) \rm Id, $$ where here $\lambda (s) $ is the fiberwise constant. Then under which condition the relative vector bundle $\mathcal E_{X/\Delta}$ admits relative Hermitian-Einstein metric? When can we get uniqueness on it? since in general $h_s=h_s'+f (s)$ and $f (s) $ is a test function .
$$\frac{\partial^2 h_t(s')}{\partial s'\partial t}=-2h_t(s')(\Lambda F_{h_t(s')}-\lambda (s')Id)$$
In fact we are facing with Hyperbolic PDE instead parabolic PDE :)
In this case $s'=\frac {1}{s}$ where $s'\to \infty$ since $s\to 0$ . In fact we have two times $t, s'$ for this deformation. In Donaldson flow we can get the canonical metric by deforming the metric by one time $t $ while in this case we have two times.
In fact the relative vector bundle is not a vector bundle near central fiber. My conjecture is that when central fiber is Kahler manifold, then we can get a stability on relative vector bundle. In fact in general the central fiber is Moishezon when general fibers are projective. Moreover the singular Hermitian metric is not well defined on the central fiber $(X_0, \mathcal E_0)$. So we need to impose some condition such that $h_0$ to be well defined. See this question
In general, I think after possible semi stable reduction, on the disc $\Delta $, the central fiber admits Hermitian-Yang Mills connection.
Hassan Jolany