流形上的泛函不等式在微分几何领域有着非常广泛和重要的应用, 因此是目前几何分析学家共同关注的热门研究课题. 本项目的研究主要有以下诸方面：首先，研究Kaehler-Ricci流意义下的几何常数、重要的泛函不等式, 其中包括Nash不等式,对数Sobolev不等式, 以及在某种指数条件限制下的紧嵌入定理;其次,基于A. Grigor’yan等数学家的工作, 研究Kaehler-Ricci flow意义下的一些泛函不等式的等价性.最后,利用上述泛函不等式,给出相应的应用,比如可以考虑Kaehler-Ricci流意义下的抛物热核,调和函数的梯度估计等等.

Functional inequalities on manifolds play important role in the field of differential geometry, and thus, they are some popular research topics which are attracted extensive attention by the geometric analysts. In this project, there are several aspects of research as follows: Firstly, we will investigate the geometric constants, functional inequalities, including Nash inequality and logarithmic Sobolev inequality, and the compact embedding theorem with certain index condition under the Kaehler-Ricci flows; Secondly, based on the works of some metheticians such as A. Grigor’yan and so on, we will consider the parabolic heat kernel under the Kaehler-Ricci flows and prove the equivalence of those functional inequalities and existence for unper bounds for the heat kernel; Lastly, we also will give some applications of those functional inequalities to the estimates for the parabolic heat kernel under the Kaehler Ricci flows and the gradient estimates for the harmonic functions.