复Finsler流形间调和映射理论的若干问题研究

This project first studies the variation of partial energy for the mapping between strongly pseudconvex complex Finsler manifolds, based on the harmonic mapping between pseudconvex complex Finsler manifolds defined as the critical point of the partial energy , so the equation is obtained,the main result of this aspect is that the mapping between strongly pseudconvex complex Finsler manifols with zero second fundamental form is no longer harmonic mapping, and the holomorphic mappings are not harmonic mapping, further studies the differential geometric properties of the harmonic complex submanifolds, and gets some basic equations on strongly complex Finsler submanifolds. Secondly, according to the volume of strongly complex Finsler manifold , we calculate the corresponding divergence formula, and introduce some concepts such as complex Laplacian operator on complex Finsler manifold, on studying Bochner technique on the complex Finsler manifold , we get are the Schwarz lemma, Liouville theorem and rigidity theorem on complex Finsler manifold, these results are generalizations of the well know results on Kaehler manifold or Hermitian manifold which were obtained by Yau and Chen Zhihua.

本项目首先研究了强拟凸复Finsler流形间映射的部分能量的变分，根据强拟凸复Finsler流形间的调和映射为能量积分的临界点，给出了调和映射所满足的方程，主要得到了复Finsler流形间第二基本形式为零的映射不再是调和映射，且全纯映射也不是调和映射，并进一步研究了调和复子流形的微分几何性质，得到调和复Finsler子流形上的一些基本方程。其次，根据复Finsler流形上的体积测度，计算出了相应的散度公式，并引入复Laplacian算子等概念，研究了复Finsler流形上的Bochner技巧，主要得到kaehler-Finsler流形上的Schwarz引理、Liouville定理及刚性定理，这些结果推广了丘成桐、陈志华等前辈在Kaehler流形和Hermitian流形上得到的相应结果。