Pseudo-Hermitian流形上的拟调和映射及其热流

This project studies the geometry and analysis of pseudoharmonic maps, p-pseudoharmonic maps and their heat flow to understand horizontal bundles and pseudo-Hermitian structures on pseudo-Hermitian manifolds. More precisely, the group transformation of Heisenberg group will be used to investigate the blow-up phenomena and the long-time existence of pseudoharmonic maps; and then the Eells-Sampson type theorem for pseudoharmonic maps will be established in more general case. The relations between p-pseudoharmonic maps and p-harmonic maps will be investigated in terms of various CR Bochner formulas. This project will also consider the long-time existence and the asymptotic behavior of p-pseudoharmonic maps. The existence theorem for p-pseudoharmonic maps will be obtained in the general case.

本项目主要研究拟调和映射、p-拟调和映射及它们热流的几何与分析性质，以此来理解pseudo-Hermitian流形上的水平丛和pseudo-Hermitian结构。本项目将利用Heisenberg群的群变换来研究拟调和热流的blow-up现象与解的长时间存在性，以期得到更广的Eells-Sampson型拟调和映射存在性定理。本项目将通过计算和估计各种CR Bochner公式来研究p-拟调和映射与p-调和映射之间的关系。在此基础上，本项目还将研究p-拟调和热流解的长时间存在性及无穷远处的渐近行为，以期得到在一般情形p-拟调和映射的存在性。

拟Hermitian几何与次黎曼几何、切触几何、Hermitian几何等等有着密切的联系。拟调和映射是水平能量泛函的临界点。项目利用拟调和热流的方法得到了拟调和映射的Eells-Sampson型存在性结果和Hartman型唯一性结果。项目还研究了拟Hermitian流形上的sub-Laplacian比较定理，并建立从完备非紧拟Hermitian流形到具有正曲率黎曼流形的拟调和映射的一类存在性。项目还研究了拟Einstein结构，推导了无迹的拟Hermitian Ricci曲率张量和Chern-Moser张量的Bochner公式，并给出了Sasakian 拟Einstein流形具有一定的刚性。