Prescribed mean curvature problems on homogeneous vector bundles

Abstract

In this paper, we investigate the existence of weak singular Hermite-Einstein structures on homogeneous holomorphic vector bundles over rational homogeneous varieties. Using Cartan's highest weight theory, we establish an explicit algebraic criterion for a homogeneous vector bundle ${\bf{E}}$ to admit a topological splitting ${\bf{E}} \cong {\bf{E}}_{0} \otimes {\bf{L}}_{0}$, where ${\bf{L}}_{0} \in {\rm{Pic}}(X)$ and $c_{1}({\bf{E}}_{0}) = 0$. When this condition is satisfied, the prescribed mean curvature equation completely decouples. By shifting the topological obstruction entirely to the line bundle ${\bf{L}}_{0}$, this splitting reduces the non-abelian prescribed mean curvature problem on ${\bf{E}}$ to Demailly's abelian theory of singular line bundle metrics. As a main application, we obtain a sufficient algebraic condition, expressed in terms of intersection numbers, under which an $L^{2}$-function can be realized as the mean curvature of a singular Hermitian structure on an irreducible homogeneous bundle. Ultimately, by overcoming the bounded curvature restrictions inherent to the classical Bando-Siu framework, this approach provides a robust mechanism to construct singular Hermitian structures accommodating prescribed singularities along analytic subvarieties.