We derive a priori 2nd-order estimates for fully nonlinear elliptic equations that depend on the gradients of solutions on compact Hermitian manifolds, which is a crucial step in solving the equations. We introduce the concept of rank of the tangent cones at infinity to level hypersurfaces of the defining function of the equation to overcome difficulties caused by presence of the gradient terms. We were motivated by the fact that there had been increasing interests in fully nonlinear PDEs from complex geometry and aimed to develop general methods to cover a wide class of equations.

中文翻译：

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Hermitian流形上具有梯度项的完全非线性椭圆方程

我们推导出完全非线性椭圆方程的先验二阶估计，该方程依赖于紧致 Hermitian 流形上解的梯度，这是求解方程的关键步骤。我们引入了在无穷远处切锥的秩的概念来水平方程的定义函数的超曲面，以克服由梯度项的存在引起的困难。我们的动机是，人们对复杂几何的完全非线性 PDE 越来越感兴趣，并旨在开发通用方法来涵盖广泛的方程组。