AI 中文总结
本文在紧致Kähler流形上研究复Hessian方程，通过约束双截曲率与解的振荡，利用插值论证而非爆破技术获得梯度估计，并应用于右端$L^1$逼近1的情形。
AI 中文摘要
在当前工作中，我们深入研究了紧致Kähler流形上的复Hessian方程。通过对双截曲率和解的振荡施加约束，我们证明了Laplacian估计可以被其梯度估计线性控制。因此，我们能够利用标准的插值论证推导出解的梯度估计，而无需采用爆破技术。作为应用，我们证明了当复Hessian方程的右端在$L^1$范数下逼近1时，即可获得所需的梯度估计。
英文摘要
In current work, we delve into the study of complex Hessian equations on compact Kähler manifolds. By imposing a constraint on both the bisectional curvature and the oscillation of the solution, we demonstrate that the Laplacian estimate can be linearly controlled by its gradient estimate. Consequently, we are able to derive gradient estimate for the solution using a standard interpolation argument, without the need to employ the blow-up technique. As an application, we demonstrate that when the right-hand side of the complex Hessian equations approximate 1 in the $L^1$ norm, the desired gradient estimate is obtained.
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