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 (Mn,g) be a complete noncompact Kähler manifold of complex dimension n≥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∈ PSH(M)∩ C0,1(M) such that ∫M(ddc u)n<+∞. 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≥3.
Related
BibTeX
@misc{dat2026mhessian,
title = {{An $m$-Hessian approach to Yau uniformization conjecture}},
author = {Truong Dinh Dat},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18171},
eprint = {2609.18171},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18171},
}
RIS
TY - UNPB
TI - An m-Hessian approach to Yau uniformization conjecture
AU - Truong Dinh Dat
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18171
UR - https://arxiv.org/abs/2609.18171
AB - 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≥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∈ PSH(M)∩ C^0,1(M) such that ∫_M(dd^c u)^n<+∞. 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≥3.
ER -