Continuity of solutions to complex Hessian equations on compact Hermitian manifolds

TL;DR

This work extends Ko\lodzieej-type $L^{\infty}$-estimates and continuity results from Monge–Ampère equations to complex Hessian equations on compact Hermitian manifolds, under the sharp Orlicz-space condition $f^{n/m}\in L^{\chi}$ with Ko\lodzieej's (Condition (K)). The authors develop a subsolution-based framework and leverage envelopes, the domination principle, and stability estimates to obtain uniform bounds, prove the existence of continuous solutions, and establish the continuity of all bounded solutions. The approach reduces Hessian problems to Monge–Ampère ones and provides a robust pathway to regularity in this non-Kähler setting, with potential implications for geometric flows and complex geometry. Overall, the paper delivers a cohesive theory for continuous solvability of complex Hessian equations with optimal density integrability on compact Hermitian manifolds.

Abstract

Let $(X,ω)$ be a compact Hermitian manifold of dimension $n$. We derive an $L^\infty$-estimate for bounded solutions to the complex $m$-th Hessian equations on $X$, assuming a positive right-hand side in the Orlicz space $L^{\frac{n}{m}}(\log L)^n(h\circ\log \circ \log L)^n$, where the associated weight satisfies Kołodziej's Condition. Building upon this estimate, we then establish the existence of continuous solutions to the complex Hessian equation under the prescribed assumptions.

Theorems & Definitions (48)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Remark 1.4
• Lemma 1.5
• Lemma 1.6: Maximum principle
• Proposition 1.7: Comparison principle
• proof
• Proposition 1.8: Domination principle
• proof