本项目计划研究具有积分Ricci曲率下界流形上的几何和拓扑。研究内容包括两个方面：积分Ricci曲率有下界、塌缩也就是单位球体积可以任意小流形上的几乎分裂定理；非正积分Ricci下界条件下的体积熵估计。在本项目中我们将通过研究具有积分Ricci下界流形列在测度Gromov-Hausdorff收敛下的极限空间的性质，特别的，观察其是否满足一定的曲率维数条件，进而来研究流形列上的几何拓扑性质。我们也将通过讨论积分Ricci下界条件下相对体积比较在任意大球上的形式研究流形上基本群增长相关问题。该课题涉及Cheeger-Colding理论、几何测度论、最优传输等领域。我们希望通过本项目的研究回答积分Ricci曲率流形上的部分基本问题，并为后续研究提供新的工具。

This proposal is concerned with the geometry and topology of manifolds with integral Ricci curvature lower bound. It contains two parts: The quantitative splitting theorem on manifolds with integral Ricci curvature bounded from below by zero and which is collapsing, i.e., the volume of a unit ball is arbitrary small; The volume entropy upper bound of compact manifolds with nonpositive integral Ricci curvature lower bound. One idea to study the quantitative splitting property and the growth of the fundamental group is to consider the measured Gromov-Hausdorff limit space of a sequence of manifolds with lower integral Ricci curvature bound and to see whether it satisfies some kind of curvature dimension condition. On the other hand, we will also study the relative volume comparison in arbitrary large balls of manifolds with lower integral Ricci curvature bound which is closely relative with the fundamental group growth. This topic roots back in several branches of modern differential geometry and metric measure spaces, such as Cheeger-Colding theory, geometric measure theory, optimal transportation and so on. This topic would solve some basic problems in manifolds with integral Ricci curvature bound and lead to some new tools in the future study.