An $m$-Hessian approach to Yau uniformization conjecture

Abstract

We develop an \(m\)-Hessian approach to the construction of finite-Monge--Amp\`ere weights on complete noncompact K\"ahler manifolds. Let \((M^n,g)\) be a complete noncompact K\"ahler 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--Amp\`ere masses of a uniformly Lipschitz plurisubharmonic exhaustion. Consequently, we construct a proper function $$ u\in PSH(M)\cap C^{0,1}(M) $$ such that $$ \int_M(dd^c u)^n<+\infty. $$ The key point is the passage from lower-order \(m\)-Hessian capacity decay to finite Monge--Amp\`ere mass, which is not a formal consequence of \(m<n\) Hessian mass estimates. We then explain how this finite-Monge--Amp\`ere 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 \(n\ge3\).