An $m$-Hessian approach to Yau uniformization conjecture

Abstract: 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\ge3) 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 $$ \int_M(dd^c u)^{n&lt;+\infty.} $$ 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\ge3).