L2延拓定理与微分Harnack不等式的相关研究

In this project, we will carry on the research on L2 extension theorem and differential Harnack inequalities. It mainly includes the following aspects: the converse of Hörmander's L2 estimate for line bundles on an n-dimensional domain; the relations between L2 extension property and Griffiths positivity；the relations between L2 extension property and Nakano positivity; differential Harnack inequality for harmonic functions on Kähler manifolds; the applications of the differential Harnack inequality for a solution of the forward conjugate heat equation on Kähler-Ricci flow. The problems mentioned above are important problems in several complex variables and complex geometry.

本项目主要研究L2延拓定理及微分Harnack不等式相关的问题，主要包括高维区域上线丛的Hörmander L2估计的逆命题；向量丛上L2延拓性质与Griffiths正性以及Nakano正性之间的关系；Kähler流形上调和函数的微分Harnack不等式；Kähler-Ricci流的正向共轭热方程解的微分Harnack不等式的应用。上述问题是多复变和复几何领域的主要问题。

本项目中我们研究了最优L2延拓定理，回答了Ohsawa关于全纯函数L2延拓的系列文章VIII中所提出的问题在一定条件下是成立的，此外我们给出了一个反例，说明一般情况下结论是不成立的；研究了Kahler流形上调和函数的矩阵型微分Harnack不等式, 给出了Kahler流形上Green函数的张量型最大值原理； 证明了Kahler流形上具有合适的曲率及体积增长的条件下调和函数的矩阵型微分Harnack不等式；研究了非线性热方程的矩阵型Li-Yau-Hamilton估计，给出Kahler流形上具有固定Kahler度量的矩阵型Li-Yau-Hamilton估计，并且给出了Kahler流形上度量随规范化的Kahler-Ricci流演化的非线性热方程的矩阵型Li-Yau--Hamilton估计，且将这些估计都推广到了约束型的情形。