An $m$-Hessian approach to Yau uniformization conjecture

We develop an $m$-Hessian approach to the construction of finite-Monge–Ampère weights on complete noncompact Kähler manifolds. Let $(M^n,g)$ be a complete noncompact Kähler manifold of complex dimension $n\ge 3$ with positive holomorphic bisectional curvature. The main new ingredient is a quantitative capacity mechanism based on lower-order complex Hessian operators. More precisely, we obtain decay estimates for suitable relative $m$-Hessian capacities on dyadic annuli and show that these estimates imply the summability of the top-degree Monge–Ampère masses of a uniformly Lipschitz plurisubharmonic exhaustion. Consequently, we construct a proper function $$u\in PSH(M)\cap C^{0,1}(M)$$ such that $${\int }_M(d{d}^{c}u{)}^n<+\mathrm{\infty }\mathrm{.}$$ The key point is the passage from lower-order $m$-Hessian capacity decay to finite Monge–Ampère mass, which is not a formal consequence of $m<n$ Hessian mass estimates. We then explain how this finite-Monge–Ampère weight fits into the weighted holomorphic-function and analytic Bezout framework for uniformization. In particular, the construction provides a higher-dimensional pluripotential-theoretic mechanism that complements recent surface results and opens a route toward uniformization under positive curvature in complex dimensions $n\ge 3$.