[论文解读] Second order estimates for equations with sums of Hessian operators on Hermitian manifolds

Weisong Dong, Ruijia Zhang|arXiv (Cornell University)|Mar 19, 2026

Geometry and complex manifolds被引用 0

一句话总结

论文在紧致 Hermitian 流形上针对自适应解的 Hessian-和方程，给出先验二阶估计，在动态极 SN（plurisubharmonic）条件下，使用复数 Hessian 运算符之和的凹性不等式实现。

ABSTRACT

In this paper, we establish an a priori second-order estimate for admissible solutions satisfying a dynamic plurisubharmonic condition to equations involving sums of Hessian operators on compact Hermitian manifolds. The estimate is derived using a concavity inequality for complex sum-of-Hessian operators.

研究动机与目标

• Motivate and study fully nonlinear complex Hessian equations on compact Hermitian manifolds.
• Establish a priori second-order estimates for admissible solutions under a dynamic plurisubharmonic condition.
• Extend existing Hessian estimate results to equations involving sums of Hessians with gradient terms.
• Handle non-convexity in gradient dependence without requiring convexity of psi in Du.

提出的方法

• Formulate the equation F(lambda)=psi(z,Du,u) with lambda the eigenvalues of the metric g relative to omega.
• Work within the admissible cone Gamma_k^(n+m) and use Real Root Hypothesis (RR) for a polynomial P(t).
• Develop a key concavity inequality (Lemma 3.1) for the complex sum-of-Hessian operator to control third-order negative terms.
• Derive and utilize complex Hessian structural formulas, including F^{p\bar{q}} and F^{p\bar{q},r\bar{s}} computations.
• Perform detailed coordinate computations using the Chern connection on Hermitian manifolds, including commutation relations and derivative estimates.
• Prove the main a priori estimate |D\bar{D}u|_ω ≤ C under hypotheses on χ, ψ, and gradient-structure g.

实验结果

研究问题

• RQ1 Can second-order estimates be obtained for equations involving sums of Hessian operators on Hermitian manifolds under dynamic plurisubharmonic conditions?
• RQ2 How does the concavity of the complex sum-of-Hessian operator contribute to control of negative higher-order terms in the estimates?
• RQ3 What role do gradient terms in the operator g play in the derivation of second-order estimates on Hermitian manifolds?
• RQ4 To what extent can existing real-variable concavity techniques be adapted to the complex Hessian setting without requiring convexity in Du?

主要发现

• Established a uniform second-order estimate |D\bar{D}u|_ω ≤ C for smooth admissible solutions under RR and dynamic plurisubharmonic condition.
• Introduced a concavity inequality (Lemma 3.1) for the complex sum-of-Hessian operator to bound negative third-order terms.
• Extended second-order estimate results to broader settings with sums of Hessians, without requiring ψ convex in Du and without concavity in g with respect to Du.
• Handled gradient-dependent terms via a careful analysis of the complex Hessian quotients and associated Jacobian/tensor calculations.
• Provided a framework compatible with Gauduchon-type equations and related geometric contexts on Hermitian manifolds.