Uniform L^∞ estimates for complex hessian equations on compact Hermitian manifolds

2026/07/26 by Truong Dinh Dat
#math.AP #math.CV #math.FA

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Abstract

We develop a pluripotential approach to complex Hessian equations on compact Hermitian manifolds. In this setting, the lack of closedness of the background metric introduces torsion terms that prevent a direct extension of the Kähler theory. Our main result is a uniform L^∞ estimate for bounded ω-m-subharmonic solutions of the equation (ω+ dd^c u)^m \wedge ω^n-m = cf ωⁿ, under the assumption that f ∈ L^p, f ≥ 0 for some p>1. The proof combines a weak comparison principle with torsion error, a capacity theory adapted to the Hermitian setting, and a nonlinear iteration scheme controlling the decay of sublevel sets. As applications, we obtain existence, stability and compactness results for weak solutions with L^p densities. These results extend several aspects of the pluripotential theory. of complex Hessian equations beyond the Kähler framework.

Citations

• Fully non-linear elliptic equations on compact Hermitian manifolds
• Complex Monge-Ampere equations and totally real submanifolds
• Complex Hessian Equations on Compact Kahler Manifold

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