定义Laplace算子和建立比较定理是Finsler几何中的难题，也是Finsler几何中的热点问题。这两个问题在实Finsler几何中已经取得突破性进展。因此，本项目拟研究复Finsler几何中的Laplace算子和比较定理，主要研究以下三个问题。第一，在复Finsler流形上引进平均Laplace算子并且研究其性质，谱问题和估计第一特征根。第二，研究复Finsler流形上的复值函数的非线性Laplace算子的第一特征根问题和特征向量光滑性问题，这里的非线性Laplace算子是由散度和梯度复合所定义。第三，在给定的曲率条件下建立复Finsler流形上的比较定理，并且得到若干相应的应用。

Defining Laplace operators and establishing comparison theorems are not only difficult problems but also hot issues in Finsler geometry. These two problems have made breakthrough progress in real Finsler geometry. Therefore, we intend to study Laplace operators and comparison theorems in complex Finsler geometry. It is mainly dedicated to the following three problems in this project. First, it is to introduce mean Laplace operators on complex Finsler manifolds, to study their proposities, spectral problems and to estimate first eigenvalues. Second, it is to study problems of first eigenvalues and smoothness of eigenvectors of nonlinear Laplace operators associated with complex valued functions on complex Finsler manifolds, which are defined by the combination of divergence and gradient. Third, comparison theorems on complex Finsler manifolds are established under given curvature conditions, and some corresponding applications are obtained.